Sneak attack always requires an appropriate weapon.
Your rephrasings to use OR and AND are not valid rephrasings of the rule. You can use that sort of rewording to make the rules easier to remember or jot them down as notes, but the rules do not include a bunch of ORs; you're making that up yourself and introducing ambiguity that wasn't originally there.
Once per turn, you can deal an extra 1d6 damage to one creature you hit with an attack if you have advantage on the attack roll. The attack must use a finesse or a ranged weapon.
You don’t need advantage on the attack roll if another enemy of the target is within 5 feet of it [etc.]
There should be no difficulty in parsing this. You need advantage and an appropriate weapon to sneak attack. Period. As a special exception, you can ignore the advantage requirement under certain circumstances. The exception is specific that "You do not need advantage on the attack roll if..." -- the exception makes no claim about weapon choice, therefore it doesn't change the rule about weapon choice.
They've gone out of their way to phrase this without the use of pronouns or subordinate clauses specifically to prevent any possible misreading, so if you need to make a ruling, it's important that you consider only the rule as it is written, and not some reworking of the text.
Another way to say this is, the exception gives us permission to replace some text in the rule. The rule says "if you have advantage on the attack roll"; the exception says "You don’t need advantage on the attack roll if..." So we can take out the phrase "you have advantage on the attack roll" and replace it with the IF from the exception, which would give us this text:
Once per turn, you can deal an extra 1d6 damage to one creature you hit with an attack if another enemy of the target is within 5 feet of it, that enemy isn’t incapacitated, and you don’t have disadvantage on the attack roll. The attack must use a finesse or a ranged weapon.
And doing that way, again, there's no avoiding the weapon requirement.
The DM can rule otherwise.
The rules as written are unambiguous about weapon choice. But if you are the DM and you really want to allow a player to do this, just go for it. DMs don't have to justify their house-rules with strained misinterpretations of the rules to find some shadow of doubt in how it's "supposed" to run. Just do it, and understand that you're operating under a house rule for the purposes of this one game. The Fun Police won't break down the door and arrest you for failing to follow the rules exactly.
Allowing unarmed (or Greataxe) sneak attacks is unlikely to break the game; this is one of those rules that seems very likely to be there for flavor reasons rather than to support balance. A sneaky, backstabby attack should be done with a traditionally sneaky, roguish weapon because you need something fast and agile to launch a sudden, unforeseen attack. It feels like the kind of thing that requires a finesse weapon.
Now, if you aren't the DM, but you're trying to get somebody on a message board to agree with you so you can wave it at your DM and say "SEE?! I'M RIGHT!" then there may be a problem at your table, because that's some pretty childish behavior.