Calling flop and turn isn't representing a flush, raising flop/turn is. When you flat call both streets you're representing overpairs/top pair, often with a good diamond in your hand. You could also have draws, but usually draws are in the muck after two pot sized bets.
I think calling the turn is a mistake. You're not even close to getting the direct odds to call, and if he has a made hand he's not bluffing a 4th diamond, he's check/folding or making a blocker bet and praying for a cheap showdown. If he has the K or Q of diamonds, you probably stack him, but he has to have one of those two specific cards and the river has to come a diamond. That situation alone doesn't get you the implied odds you need to make this call. As a general rule, you don't want to be relying on the implied odds of hitting a flush with 4 of the same suit on the board.
Shoving the turn is also a mistake, for the reasons you indicated. You're getting very few better hands to fold. Folding is probably your best play.
When you bet the river, you're polarized. The one pair hands that you were representing by calling the flop and turn are checking behind, so when you don't do so you represent either a slowplayed flush or a missed draw. You can't really have any sets/two pairs/straights, because those hands are going to be playing the flop and turn a lot more aggressively with a 3-flush on board.
If you're going to bluff the river, don't bet small. I'm probably not folding anything better than TPGK to that bet. Sometimes you'll have made flushes, but if you have a missed draw more than 25% of the time there the call is correct. There's a reason being polarized and overbetting goes hand in hand. You need to be shoving the river with both your missed draws and your flushes. If you think you're getting a competent player off a good pair or better giving him 3:1 on the river when every draw missed, you need to reevaluate.
Given your notes about villain double barreling with air, and the fact that he folded to a small river bet, chances are he had garbage, or one of many missed draws. KT, QT, KQ, AT, Q9, and K9 all with a diamond I think are likely hands for villain. I think checking back your showdown value is the best play on the river.
Imo you got lucky that you were able to take down the pot on the river. I think in the long run both calling the turn and betting small on the river are both -EV in comparison to folding and checking behind.