I was betting for value the fish/maniac, and then... Did I play it correctly? Why would he check first the river after calling other streets... To induce a bluff? But would a player with a VPIP of 64 be able to do this or did he simply bluff me out with a missed flush draw?

PokerStars - $0.02 NL - Holdem - 7 players
Hand converted by PokerTracker 4

BB: 125.5 BB (VPIP: 10.15, PFR: 5.84, 3Bet Preflop: 0.00, Hands: 396)
UTG: 97.5 BB (VPIP: 11.96, PFR: 10.87, 3Bet Preflop: 5.41, Hands: 95)
UTG+1: 54.5 BB (VPIP: 10.00, PFR: 10.00, 3Bet Preflop: 9.09, Hands: 21)
MP: 101.5 BB (VPIP: 61.11, PFR: 16.67, 3Bet Preflop: 10.00, Hands: 37)
CO: 71 BB (VPIP: 17.86, PFR: 0.00, 3Bet Preflop: 0.00, Hands: 28)
Hero (BTN): 114 BB
SB: 100.5 BB (VPIP: 40.00, PFR: 20.00, 3Bet Preflop: 0.00, Hands: 16)

SB posts SB 0.5 BB, BB posts BB 1 BB

Pre Flop: (pot: 1.5 BB) Hero has J A

fold, fold, MP calls 1 BB, fold, Hero raises to 4 BB, fold, BB calls 3 BB, MP calls 3 BB

Flop: (12.5 BB, 3 players) 2 T A
BB checks, MP checks, Hero bets 6 BB, fold, MP calls 6 BB

Turn: (24.5 BB, 2 players) 8
MP checks, Hero bets 11.5 BB, MP calls 11.5 BB

River: (47.5 BB, 2 players) 7
MP checks, Hero bets 15.5 BB, MP raises to 80 BB and is all-in, fold

MP wins 76 BB

bigger turn
wp otherwise

Preflop
I would raise a little bigger to 5bb. Maybe even 6bb. You want to get as much value as possible, and you want to isolate and go heads up to the flop. In smaller stake games where opponents are super loose pre, this is often a good idea. To get a sense of how loose opponents are pre, think about how many pots go multiway to the flop.

Flop
I like a bigger bet on the flop. A half pot bet gives flush draws odds to call against you, especially given their implied odds. Tx and 2x probably don't have odds to call, but it's close, depending on the implied odds those hands have on you. Ax you're crushing, but Ax would be willing to call bigger for sure. You want to bet a size that forces your opponents into mistakes, and a smaller half pot bet like this doesn't accomplish that goal.

One consideration is that if you're cbet bluffing you probably want to go smaller like this, and forking your bet sizing could let your opponents know "oh, he has value when he goes bigger, and bluffs when he goes smaller". At NL2, I just wouldn't worry about this. Even at higher stakes.

Turn
My thoughts are similar to my thoughts on the flop, although with only one card to come, draws don't have as good odds to call. Here, I just think that you're missing value by going smaller. I think that flush draws and Ax are going to call a bigger bet, and Tx very well might as well. Another consideration is that you want to build the pot so that you get more value on the river versus Ax. See Progressive Bet Sizing.

River
My first thought that this was a nasty river card because it's hard to get value from a T or an 8, but there's a good chance that you aren't getting value from that anyway for a third street. I think the goal here should be to seek value from weaker aces, and hope that Villain also might call with JJ or JT. The weaker aces would call something like a 2/3 pot bet here, and I think that there's a decent chance that JJ and JT do as well at NL2, so I would definitely go bigger. Sizing it smaller like that is targeting something like 98 or 76, but I just don't think that those combos are a big enough portion of Villain's range.

Facing the raise, awesome job getting out of the way. This is a perfect example of a spot where you should be bet-folding. There's no way Villain is check-raising something like A9 for value that you beat. And that means that your AJ unfortunately becomes a bluffcatcher. Which means that Villain needs to be bluffing about 30% of the time for you to call. People just don't bluff enough in these spots to justify calling until you get up to the mid and higher stakes. Check out Stop Paying Them Off.

This