Both 386 and 357 cells are 1.5V, and will theoretically run the watch perfectly well. The 357s will last longer because they have a larger capacity.
The fact that one of the cells got hot, and the display dimmed almost certainly indicates that the cell had its terminals shorted. In other words, the positive and negative terminals were bridged to each other by metal and delivered a large amount of current (through that piece of metal) in a very short time. It would have very quickly depleted the cell.
I'd say the 386 moved around (because it's much thinner) and the butterfly clip probably bridged both terminals.
BTW, you can easily tell how a battery is to be inserted. If the bottom of the battery well is bevelled, then it's negative down. If it's a 90 degree transition, then it's positive down. On the P4, it's also engraved on the plastic carrier
Rgds.