First, @
kbrakke , have some XP. This is thoughtful and evocative thread on a fascinating, and obviously myriad, topic. So, kudos to you on that.
So to further my understanding of this game and perhaps help others see different aspects of this wonderful game we all presumably love, I have some questions.
1- What does balance mean to you?
"Balance," to me, is some gamer/game designer buzzword jargon. Like other subcultural jargon, it is particularly ill-defined and/or has multiple definitions depending on who you ask, and hold, essentially no meaning outside of the users of that industry/subculture. i.e. It's a "made up term" with a "made up definition" that people want to use and treat as incredibly "serious business" in the "gaming" or "RPG" subculture.
For the most part, it seems to be used as both attack and defense by those who presume there to be a mathematical or "objective equality" between options and choices within the game. Feat to feat. Class to class. Race to race. Subclass feature to subclass feature. {perhaps most importantly to most] "Damage per Round" to "Damage per round." Everything should be "equally good" or, at least, not "noticeably worse" than something else.
A "belief," since really that's all that it is [not some objective truth], that the numbers (i.e. how "power" is generally marked and tracked in RPGs and other games) must be "balanced." If these options aren't ["properly"] "balanced," then somehow, someone is not capable of having as much "fun" as another, or isn't capable of "fulfilling their role" within the party adequately, or generally succeeding in [if not some misguided concept of "winning"] a given encounter or the entire game.
This is, of course, nonsense and objectively untrue [for D&D, if not all TTRPGs]. Yet, people will get into very lengthy and/or heated debate over the presence or lack thereof of this perceived industry buzzword, "balance," which up to a couple of decades ago, no one would ever hear at a D&D table.
I would say, at it's simplest and most easily adopted use, it would mean, "fairness." But "fairness" is not at all dependent on a strict accounting of numerical benefit or equal powers. The mythological fallacy of "Linear Fighter/Quadratic Wizard" was never a question of "balance." The Fighter had abilities and powers that were [vastly] superior for success at lower levels, as compared to the M-U/Mage. The Mages, then, gained vastly superior abilities and powers at higher levels. The classes were designed to be "balanced" more of less "fairly" for over the length of a campaign, as the game was written and presumed to be experienced. They "evened each other out," as it were.
Have there been things in D&D's history that have been obviously "unbalanced," a.k.a. over- or under-powered or blatantly unfair/uneven? Absolutely. But "balance" as a core design concept is largely a fabricated construct of the industry that people now use and toss aaround as if it is some gospel objective truth.
And, simply, it is not.
2 - When you balance an encounter, what is your desired outcome for that encounter? Do you balance combat on the encounter level, the adventuring day level, or the campaign level?
I don't "balance" encounters. I generate an encounter based on the story and internal consistency of the game world. Yes, of course, I try to keep things, generally, manageable much of the time. You're going to see giant rats and skeletons at low levels. But if a those skeletons are minions of a necromantic lich-dragon that is waaay out of your league, and you don't take the cues or clues [or outright proclamations] seriously and try to engage at 2nd level, then, yeah, you're probably going to get TPK'd unless you have enough sense to run away or otherwise try to talk your way out of things.
The "day level?" Like, oh you're running low on spells so we should just let you take a break here...when you've just completed a heavy combat in the midst of an enemy fortress, the alarm has been raised, and everyone is looking for you? "Sure, yeah, here's a quiet room no one's going to find you in for an hour. Can I get you some tea? There's some nice hot tea in the room too. Blanket? Sure." That's not happening.
"Campaign level" I am taking to mean, like I said, "am I going to throw generally level appropriate stuff (enemies, traps/hazards, environmental factors, etc etc...) for the party's general level?" The answer there is, "Over the course of the campaign? Generally, yes."
It may be possible to design encounters in this way, but it is certainly not how I approach creating an encounter or adventure (and definitely not a campaign). One of the key elements -if not THE key element- of a successful D&D game is that the group/table (including the DM) is generating this story together. What the characters DO is what leads to what happens and where they end up. What they do and where they end up do not ebb and flow to suit what they are capable of handling!
They can, sure. Encounters can just be made up on the fly or things altered if stuff is getting slow/boring for too long...that's part of my job as DM...But it not something that can be "deisgned/planned for"...because what the characters are going to do, where they'll go, how they will CHOOSE to use the features and treasures and magic at their disposal is completely up to them.
How the elements of the world respond to those doings, goings, and choices, that's how up to me[the DM], and how the campaign happens. Not by "balancing encounters."
3 - When you look at your players/other players what things make you feel like something is not balanced?
If things are obviously "unfair" or "uneven" in a given confrontation -innumerable enemies, some skill challenge that [I, as DM know is] set to 50, one class gaining a feature that allows them 5d10 additional damage on attacks that no one else gets- then they could be said to be "unbalanced" (or as some here are preferring, "imbalanced"). But, then, if a PC has a magic weapon that other PCs don't have, and that weapon let's them do extra damage other PCs can't match...is that "unbalanced?" Is it even "unfair?" As a DM I have NO compulsion whatsoever to say, "Well now I need to make sure everyone else gets a magic item or device that let's them deal an extra xdX so we're all even." That's just part of the game, the results of players' choices, and how the fates have swung.
So, I'm not really sure how to answer this...I suppose the best/closest I could come up with would be to use another industry/subcultural jargon term..."broken." If something is obviously, objectively, "broken" then I would/we can say it is "unbalanced."
4 - If you claim that you do not worry about balance in your encounters, what are your overall desired outcomes from combats?
My who what now?
I don't have "desired outcomes" for combats.

I mean, as a matter of "fun" for the table, my HOPE is that things will go well and the characters will succeed in whatever the encounter is. Live to fight another day, topple another evil, and all of that. But I have no -and certainly don't "design" for- any desired outcome of a combat...again, other than as is befitting the situation and internal consistency of the game world.
I mean, I'll -again in a general sense- think to myself, "This encounter/trap/interaction/whatever it is should be simple[/medium/hard]." or "This climactic endgame battle with the BBEG should be an amazing cinematic throwdown masterpiece!"...Whether either of those things ever ends up being close to true, however, is in the players' hands, dependent on their choices and the actions of their characters.
5 - If something seems imbalanced to you, how do you go about fixing it?
I'm not sure what you mean by "fix it."
If you mean for something like a class option or something from the book? I'd probably "just say no." (child of the 80's and all) and not allow it into the game.
For a clearly unevenly matched combat situation gone awry or a player having a "bad dice" night whose beloved longtime character is on the brink of death, an extremely risky attempt at a nigh-impossible skill check or somesuch? I'm not sure there'd be much I could do, other than mitigate the foes/reactions to be more lenient on the PCs, have them be captured or try to force a surrender or something...
Ultimately, again, this is all in the realm of player[PC] choices and actions...the outcomes of most- if not all- situations, "unbalanced" or not, will be decided by them.
6 - In video games or card games something is considered balanced if it has an overall 50% win rate against the field. A character in a fighting game would be imbalanced if it consistently won more than half its matches. Or a deck in Magic would be OP if it was more than 50% to beat the field. In dungeons and dragons that sounds absurd. My parties are probably around 100% win rate. Do either of these numbers make sense to you? Would you play in a game where the players "won" half the time? What does that mean to you?
"Do either of these numbers make sense to [me]?" Not in the slightest. A game with 100% success is no a game. What is the point? You know you'll succeed in all things, every time. Where's the challenge? Where's the -what many point to, rightly, as the core point of playing at all- fun? "Oh look another dead dragon and treasure horde. Gold star for us. We all level up together! Yay!"
I don't know if "50%" is a realistic expectation for D&D. I certain hope a party does better than 50%...but the crux and difficulty in this question is, again, jargon. What do you mean,
"...where the players 'won' half the time?"
Do you mean "won" as in they lived through it? "Won," as in they succeeded in the given task or goal of the overall mission/adventure? "Won," as in found the treasure and didn't have to kill anyone? "Won," as in killed all of the monsters but didn't ALL die, themselves? "Won," as in none of them died?...or didn't have to expend significant resources? ..or?...or?...or?
If we run away, and no one died, did we win? If we try to RP haggle, but still (either from lousy roleplaying or a flubbed dice roll) pay full price, did we win? Do they count toward my 50%?
This is something of a meaningless question without an answer. I would certainly hope any group I played would succeed more than 50% of the time. But I have no desire to play in a group where success is a foregone conclusion (100%).
"What does that mean to [me]?" Absolutely nothing.
As a rule, I think trying to apply "video game" numbers/statistics to anything related to D&D/TTRPGs or "games/gaming" in general is a fool's errand. It produces mistaken and unrealistic expectations and is not particularly useful in any respect.