When pushing the gas peddle for the first time...

I agree it may be the fuel pump, but there may be more than one filter to change! A car that age should still have a good pump, but you never know. Clogged fuel injectors can give the same symptoms as well. My first step after double and triple checking for a second filter would be to add a bottle or two of injector cleaner - I like Techron, it works well. See if that helps.

Simple fuel pump check would be to disconnect the fuel line at the engine- while the engine is cold, preferably after it has sat all night. then stuff the end of the hose into a bottle (I use a 2 litre lol) and observe the rate of flow. If it looks normal, you still may have a problem with pressure. Many auto shops can check your fuel pressure for you for not much $$. I d have them do the checking, unless you have a buddy with a fuel pressure gauge.

Also...

I know in 1990 Mazda still used a disributor cap and rotor.

I can't find any info in my car library that tells if it does still in 1991.

If so, replace them. Double check all wiring to the coil pack if it does't use an old fashioned distributor.

My bet is that it't injector or fuel pump related, but those are other things to look for. Best of luck! Tom