What are binding bets in live poker?

In general it will depend on how the room handles it and it might also depend on past cases that you were involved in (which in all likelihood has never happened before)

In rooms that I played in typically if 3 people heard what you said then it will be binding. So in the 1st case where the entire table has heard you say call then in my experience it will be binding. Except one time where all the players and the Dealer heard what the player said and I asked the dealer if the player was all in and the dealer said yes and then the Floor got called over and a friend of the player said out loud that the player hadn't said anything and the player claimed he never said anything and the Floor didn't ask any other player or the dealer if they heard anything. I stopped playing in that room for a while because the Tournament Director refused to come over to the table to investigate the obvious cheating that had happened and the horrible decision by the Floor.

In the 2nd case assuming that you immediately responded to the dealer's mistake then you will be allowed to act. The player who exposed their cards will not be held to any penalty if a penalty would normally apply for exposing their cards (like in a tournament). However if several other players also claim that you said "call" it will be a tough decision for the Floor. I have seen something like this happen when the dealer thinks that something said at a nearby table was what you said. But I have never seen a player (including myself) held to a Dealer mistake like this.

Since I don't play PLO I have no idea if "half pot" or "a third pot" are legit bets. I would guess they aren't but then what do I know?

That's interesting, because in my (limited) experience playing live, I've always relied on what the dealer said when I wasn't completely sure. But based on what you're saying (if the dealer misheard, then the dealer's mistake isn't binding), I would really need to double check with the rest of the table: "Do at least 2 other players agree with the dealer about the action?" If the rest of the table doesn't speak up, then the player can always claim the dealer misheard? This seems like an obvious loophole if the table is short-handed lol. If we don't trust the dealer, it's impossible for any action to be binding.

Dealers are human and will make mistakes. In most rooms the Floors will defend player's rights and action. And yes in this spot you should check with the rest of the table but first you have to immediately say that you did not say "call".

The thing that helps me in these situations is that I don't talk when its my turn to bet. I don't say "call" and I don't say "raise" unless I don't have chips that can do the talking for me. So part of the argument that helps me is that if the dealer thinks he heard me say "call" part of my explanation to the Floor would be "I don't ever say call I just put one chip (or more) out there."

In fairness to reality I do sometimes have to say "all in" if I have a lot of stacks of chips that I can't put out there all at once. One time I asked the dealer for his all in button before I did or said anything else. He hesitated but then I asked again and he handed it to me and I put it in front of me and I said "all in" and then moved all my stacks up to it.

There are times when I have to announce my raise size like if I don't have any hundreds and I want to raise to 600. But that is life. In terms of calling though I never say it out loud (at least not until after I put a chip out there).

1 and 2 are floor calls and they will try to make the best of the available information. 1 is probably a call and 2 is probably not, but things can get weird if there's conflicting information.

For 3, saying "pot" is only binding if it's a pot limit game. For any other game or statement such as "half pot" the player will be forced to bet SOMETHING. Action stays on them until they put chips in or state the amount.

Verbal is generally binding (with the exception of non-binding statements such as #3 as reducto explained above). It's typically going to come down to the floor's discretion as to what was actually said.

In practice though if you say one thing and the dealer mishears you you should IMMEDIATELY speak up and correct them. Typically they're going to rule in your favor in that scenario. Actually the floor will rarely even be called, the dealer will just correct the action and the hand will continue.

When there's a problem is when a player states something then there is additional action or another player reveals their cards. In that case it's much more likely that the floor will rule that whatever the other players thought the action should be stands.

verbal is stronger than chipspeak.