Warning: 6-max donkey grunch.

To be aggressive on the turn against a somewhat passive player or not pairing your A. I think that the second one is better. Personally, I may have played the hand like first since I'm new to 6-max and have been pressing the pedel too hard on more than one occasion.

If BB had lead out on the flop, I'd raise the turn and play it like hand number 1. Combine that with his passivity.
It looks like he hit either a stronger A than you (i.e. better kicker) or perhaps a raggedy 2 pair. But depending on BB's per street aggression tendencies, he may like to pop on the turn w/ sets (or maybe 2 pair). Unfortunately, 35 hands is a pretty small sample to determine how accurate these stats are. But if you've seen him pop once or twice, I would think that this would increase your confidence in this greatly.

It sems that hand number 2 has higher EV than the second in the long run.

Also, I think the other streets as they were played are fine. On hand number 1, it looks like you are behind and you didn't improve on the river. The second hand, a call down is fine.