I agree, some details were skipped!
I don't have an exact relay model but we are talking about signal type relays with a 4.5V or maybe 5V coils with a low current draw, 1 coil latching.
Both coils should switch at the same time, and your description is accurate, I want a single push of the SPST tactile to latch them and the second one to unlatch them.
I thought about this a little bit and I remember something called a multiplexer from logic class in university but I admit I forgot almost everything by now but maybe something like that could assist here?