If you played in games where the edges were more thin and ranges were wider, you might see people correctly call when pricing in their backdoor draws. Did the villain have a single ? If so, folding might be pretty bad. If he can discount AK and KQs from your range and he'd holding the A of (blocking your pair + A high FD), he could think he has a BDFD, gut shot, and his A is good some % of the time. Even the J could add enough equity and be a blocker. You see this kind of thinking from online shorthanded players, where you're looking for perfect 3 straights, backdoor draws, and maybe hitting 2nd or 3rd pair being good. Shorthanded players are sticky. That's the best aspect of a hidden hand that improves to better than they're drawing to. They improve and lose.

Back to our "protect the hand" thing... if one guy has AJ and one guy has J9. They both improve on a T Every $ J9 puts in has 0 equity.

Yeah, the range, vs. range, what range can villian expect leader to have is a bit tougher. I mean, he has like KJ, KQ, 77, JT, or a heart draw, and basically nothing else. I don't even like leading with any of those, except maybe KJ. But, against standard low-limit opponents I am basically always going for a c/r on this flop, which is I'm sure exploitable by a stronger GTO opponent.

Pot's multiway so has to contend with the FD too, so really has only ~4 outs against this range, and potential flush draw in my guestimation.

Agree the "protect your hand" is not very applicable here, except as a fallacy. I'm betting the J9 calls two-bets here too.

Just to be clear, if I thought the BTN would spite raise our donk or thin value raise it, I like 3 bets more than 2. It is also more likely to trap the field on the installment plan. The reason I c/r is because I think a bet is way more likely than a raise, thus 2 bets is better than 1. We assume stuff about the villains and make plays based on that. Change the assumption and we do different stuff.

This is a good way to look at it. By check raising, you make that mistake more expensive with the exceptions noted above. Also note that check 3 betting makes their mistake of calling three bets cold an even bigger mistake. You're giving your the opportunity to either fold their equity or pay to realize that equity. Great no limit holdem players are especially good at knowing just how much to bet in order to either get a call or a fold. We don't have the option of mixing our betsizes in limit holdem, but we can make our opponent's mistakes more expensive by getting them to put in the most possible bets when we have big hands.
I'm not really a fan of putting people on specific hands, unless ranges actually get so narrow that I think they have exactly one hand. For example, I was playing uncapped limit holdem online and made a set with QQ on the KQ2r board. No flush by the river and of course my opponent wants to keep raising. I reraised a few times but eventually called expecting to see KK the majority of the time. Yup he had KK.

Another time my opponent cold capped the A63 J turncard when I had 33. Ace on the river and I just call totally expecting to see AJ. Yup he had AJ.

In these spots, my opponent's ranges had become so narrow that I could put them on specific hands. Without the action to tell me that I was beat, my hands were monsters.

Usually, you can't be so sure as to what your opponent is holding. It would be nice if we all could think about ranges fluently at game speed, but realistically, many of us can't. If this is the case and you need to focus on a specific hand, I suggest thinking about where your opponent puts the "good hand/bad hand" threshold. For example, if I was on the button I would put your "good hand/bad hand" threshold somewhere around KTs for betting this flop. Yup, I put you on KTs and anything better, maybe throw in the occasional JTs for the open ended straight draw. I think you would check anything worse. Then when the action comes to me, I'm gonna estimate my equity vs your betting range. I'm gonna raise when I think I have enough equity to jam and I'm gonna call when I think I have enough equity to bluffcatch and draw. I'll fold if I don't think I have enough equity to do either.

I would donk this flop because of our relative position and because it's a kill pot. The PFR is the button and thus the most likely bettor and if you start with a check you are likely facing the whole field with having to call $16 cold. That's not really ideal for getting more money in the pot. I would donk and give them a chance to call for one bet and hope the PFR has a hand he can raise with. If this was a normal 4/8 pot, I don't mind check-raising as much because they are less likely to fold. Yes, the pot odds are the same, but these people aren't thinking about pot odds, they are thinking $16 is twice as much as $8 and their calling ranges will tighten up.

Some of these guys are talking about what your hand looks like (a monster) or that certain players will just call down with TPTK based on the strength of your donk, but this is a 4/8 game and I find that the vast majority of players at that level are happy to spew here and that there isn't a lot of thinking going on, in general. I would just take the line that is likely to get the most money in the pot and not worry about how they might interpret your range.