The HAI model numbers are not on the UL sticker. They are all next to the hand written 1.
1 - OPII board with two expansion daughter boards
HAI 10A06-1 16 Zone hardwire expander daughter board. Under could be another one or as you say a serial board.
2 - HAI 17A00-8 16 port zone and output board (middle)
3 - HAI 19A01-1 HAI 8 relay module (bottom) - many folks here use this device for their irrigation system.
Lots of connections as it looks to be that you are at port capacity fo all of the boards installed. IE: no blank screw terminals that I can tell. 16 + 16+ 16 (48) zones and 16 outputs and 8 relays.
Do you also have an Omnistat Thermostat connected (it'll work without the panel), keypads, touchscreens, audio et al?
Note that all of these boards are interconnected. The smoke came from the top board so it is most likely damaged. You do not really know the condition of the other boards right now.
Personally with 5 HAI boards it might be a bit difficult to DIY fix this or it could be just the main OmniPro 2 board on top that needs fixing making it easy. The terminal strips on the OPII board just lift up - same as the two daughter boards. IE: you can just remove the stuff and slip the OPII board out from underneath.
I would first unplug the battery / both transformers as right now you have a disfunctional brick and leaving the power on this stuff might damage it more.
Give a call to Leviton/HAI and Automated Outlet and tell them what happened and show them a picture of what you have.
Where are you? East, midwest, southwest, west, northwest?