And so I have yet again unintentionally demonstrated that it is virtually impossible to make any statements on the internet that will be understood the same by everyone.
Rather than attempting the impossible and trying to rephrase the options so that everybody has one that fits, I'll just clarify individually which meaning is intended to fit certain answers. (So far
@kenada is the only one for whom "Other" is really necessary.)
I have to go with "other," because I exclusively run Original/Basic D&D, and I don't want players bringing a concept to the table before they roll their stats (3d6 in order). If I'm running a fantasy campaign, the playable choices are likely to be fighter, magic-user, cleric, thief, and (depending on the setting!) possibly elf (fighter/magic-user), dwarf (fighter), and halfling (fighter) as well. If you come to the table dead set on playing, say, a magic-user with a particular tragic backstory, you'd better hope the dice are kind and give you a decent Intelligence score.
I didn't explicitly mention whether players should be making characters (or getting attached to concepts) before they meet up with the group because I didn't want to muddy the waters with that particular dimension. The questions are intended to be inclusive of that preference.
In this case, the answer designed for your explanation is either #3 or #4, depending on whether you want the players to turn the results of those dice into an idea that seems like it was made for the fiction of the setting and campaign (#4), or whether you're fine with them coming up with something a bit outside the box as long as they can make it fit pretty well (#3). My guess for an OD&D/BECMI 3d6 stats in order would be that you're going for #3.
I went for option 3, but I'm really unsure of the distinction between that and option 2. Are the limitations of "Must be able to be translatable into the setting" (generally Eberron) and "Must be able to work with the rest of the party on the adventure" enough to push option 2 to option 3?
Yes if you are sticking with the options allowed for Eberron (I acknowledge that the 3e version of that was "pretty much anything", though the game has changed enough that I'm unsure whether I would run it that way in 5e) and working with the party means working
well with the party. No if you mean that any option from an official game source automatically is approved regardless of how much work it is to fit in, and working with the party just means not destroying the campaign.
Other, as both player and DM: Freedom within limitations.
Limitations: the DM sets, or is able to set, some setting-wide race and-or class limitations* e.g. "no Dragonborn", or "no Monks", or "core five races only", or "Necromancers must be Evil".
Freedom: within those limitations, anything goes. "Yes Necromancers must be Evil but you're free to play one if you like". "'It's what my character would do' is a valid explanation for anything provided that it more or less fits in with what your character would do based on its established patters/personality/etc.". Group cohesion is nice but it's by no means demanded or enforced.
The only other meta-limitation is that what happens in character stays in character and doesn't extend to the real-world table.
* - which can also be expansions e.g. "for this setting we'll be using I've designed a new playable 'Pirate' class".
That fits in option #3, because it establishes a tone for the world and the limitations are in service to that. None of those excluded options would "break the game", but they might very well interfere with the intended play experience for the setting and campaign. It's a good subtlety you bring up that deserves further explanation.
Basically, #3 can accommodate themes as simple as "do whatever you want, as long as it's fantasy and PHB only" as long as the reason for that is because of a desired setting and campaign experience, while #2 would never make such broad limitations unless the DM really believed that a dash of sci-fi would completely destroy what had been previously established, and everything outside of the PHB is mechanically broken.
Gods, this argument again?! This is a provable false choice.
We factually know that both Forgotten Realms, a kitchen sink setting that contains all of the options, and Dark Sun, a setting that eliminates a number of racial and class options, sell well enough to be redone for multiple editions.
So we know, factually, that both all-inclusive and thematically limiting settings are popular. Any attempt to pick a "right balance" isn't correct. Just like there are more then one type of spaghetti sauce per brand, the idea of a "right balance" DOES NOT EXIST, rather there are multiple valid "local maxima" - lots of points where different tables fit at different times. Averaging them together does not find you a "right balance", and can easily not even be one of the peaks and can instead be a compromise no one likes.
It's such a re-tread, tired argument that is fundamentally flawed.
It's not a topic I'm making an argument for, it's a preference poll so I can determine how popular the various choices are. Text discussion of these sorts of things alone can't always provide as good of a perspective on the popularity of choices. Does relative prominence of preference matter for your table? No if you have an established group. Yes, if you want to have a better perspective about what new players are likely to expect. (Well, technically I did make one claim--that DM-Overseen campaigns are a valid style, and provided an option in the answers for those who disagree with that claim.)
