"Campaign", as I've always understood/used it, simply meant what a particular group of PCs [at least to start, replacements and additions are a given expectation] was going to do.
It could be/cover any amount of in-game space, any setting, scope, number of DMs, tone, or even out-of-game time...they're all "campaigns." Most times, there is no set objective or expectation of levels covered beyond a general "starting level." To whit...
"I'm taking a break and Jim wants to DM/has a plot idea/has wanted to DM for a while. Everyone roll up 1st level characters." How far are we going to get? What the overarching plot actually is? How long/many months will we be playing this? None of that matters. What the setting is, restrictions/additions to character options, Houserules we will/won't use? That's all expected to be known/shared prior to play beginning. But that group of [starting] characters with Jim as the DM, that's a campaign. Likely to be referred to as "Jim's Campaign" when spoken about among the players...to differentiate from "My Campaign" or "The Desert [setting/world/characters] Campaign."
"Hey! Let's do a Dragonlance game. Here's the pregen/book characters or [here's the setting spec's] come up with your own. Figure 4th level to start and we're going to run through/follow DL1-10...see how far we get/where that takes us. Though Chris might make some tweaks along the way so you can't really go strictly by how the books went." That's a campaign.
"I wanna play an Underdark adventure. Tommy loves the underdark and has a thousand ideas for a high-level/power adventure he's been working on for while. So we're all set! Everyone grab a PC you already have or come up with a new one, no lower than 10th level...if you want to live past the first session. Bwahaha." Whether this is purely Tommy's made up stuff, using all/only official Forgotten Realms Underdark/Drow stuff, or running us through D1-3 to get us to Q1, doesn't really matter. That's a campaign.
"I'm tired guys. Could be take a break from the big overarching plots and world-saving? We just do a simple roll up. Start at 1st (or 4th or 8th or wherever you want) level and have a good old-fashioned dungeon delve one-shot?" YOU SURE CAN! That's a campaign. If everyone has a great time that session and wants to continue? Fine. If someone really falls in love with one of those characters and wants to use it in some more consistent game, that's fine too. Take that one-shot character and dream up what they're like at 10th level to use in Tommy's underdark game, that's cool too. But even if they are all played, slain or forgotten, that night of that one-shot dungeon adventure, that was [at the least the potential beginning of] a campaign too.
"DUDES! I just bought the "Railroadallinarow in Onehardcoverbook Adventure Path"! Levels 1-25! Looks awesome and should keep us busy for the next 9.25 months [if we follow it exactly/level when it says]." That's a campaign too. Not one I would necessarily care to be in...but it is a campaign.
A "module" can be one piece of a campaign or the entirety of it or, to break things down further, just one piece of an "adventure."
There can be multiple adventures within a campaign.
An "Adventure Path" is a whole [probably single] campaign broken into multiple adventures/modules in a box.
A "game" is synonymous with a "session." A game/session is, likely, only a part/fraction of a particular module or adventure, and thus a singular [probably small] moment of a campaign...unless the campaign in question is a one-shot in which case that game/session is the whole campaign.
That's how I look at it anyway.
