I think you should go talk with those masajid imams about this issue too that why is this time difference. There can be one reason which i can understand is the jaafri fiqh(or shia) iftar 10 minutes late. But i don;t think azaan usually take 10 minutes anyway.
Basically you are not supposed to iftar on azaan, but on timing calculated by credible sources for ...
Saying "Bismillah بسم الله " is not mandatory, but among the sunan of eating.
This is based on:
"When one of you eats food, then let him say: 'Bismillah.' If he forgets in the beginning, then let him say: 'Bismillah Fi Awwalihi Wa Akhirih (In the Name of Allah in its beginning and its end.)"
It is also permissible to ...
I suspect this is just a practical constraint.
If we exercise during daylight, it may be too hot, and we can't drink to rehydrate until iftar. We may also need to go to work.
It might be impractical to exercise at night. E.g. the gyms are closed, people tend to sleep at night. Moreover, eating a large meal at iftar can make it hard to exercise ...