Implied odds I usually consider as calling a small bet that will lead to several times the amount of the bet when I hit a big hand.

You were calling $10 (without odds) on the flop, and he only had $15 left in his stack. You called $9 (with odds) on the turn, but he only had $6 left in his stack.

If you think you have 16 outs and you know you're going to play it to the river, push after his preflop raise to $10. In addition to your apparent 57% chance to draw to a winning hand, you add folding equity. Once the turn card hit, you are going to lose that hand more than win it, even though you have pot odds to call. It's far better to get your money in with the (apparent) best of it than to wait until you have only a 13/46 chance to win.

The way you played it, you gave yourself a chance to save $6 if your draw didn't hit.

Edit: It's actually 15 or 12 outs since one of the 3's is a diamond, so you didn't have odds (32:15) to see the turn even if you thought the A's were outs. The small implied odds would make it a marginal call if you thought he would bet or call if a third diamond showed on the turn.