Where do I caulk on the outside of leaky windows?
[I] noticed there were gaps in the caulk between the board trim (see image below) and the window, so I re-caulked all the windows


Gaps between the board and trim are normal. The standard way this is done is you have your siding in layers


Siding board (OSB)
House wrap (plastic or tar paper)
Window tar flashing (bottom)
Window
Tar flashing (sides and top, hopefully nested just under the wrap)
Siding


Caulk can improve on this some, but properly installed windows like this should not need much, if any, caulk. Any water that gets behind your siding should be channeled down to the ground.

My bet is the window was installed incorrectly. A common area builders like to cheat on is wrapping the house wrap over the edges of the sill. That can create a pathway for water to penetrate the house. Caulk can't fix that either, since water is getting behind your siding.

Your best bet is to pull the siding off next to the window and make sure it's layered properly. Check the bottom corners if you can, since you typically need extra wrap there.