MTQRL and UMRA are 3/4 midgets or TQ's. They have be around for 60 years or so. They are powered by mostly by de-transed motorcycle engines. They are racing purist, they believe that a true race car must be shaft driven to a quick change rear end. They didn't feel this way until the 1970,s. They are mini-midgets and require mostly special TQ components.
AMSA and MMSA are mini sprints. They are in the middle of an an identity crisis. They began back in the 1980,s when somebody took an old midget frame and installed a motorcycle engine with the trans and used chain drive to the rear axil. They also installed a wing because they were world of outlaw wanabe's. I can still remember the 4 wide parade laps. As mini sprints grew a bunch of new people decided that "mini" was too mini and did not match the ability of these cars. A guy up in Canada said he thought Lightning Sprint was better, so now we have minisprint-lightning sprint deals some groups even use the two terms together.
Mini sprints are still midget frame with late model motorcycle engins in them. Several groups have decided to make the drive optional either chain or Quick change. Mini sprints can be 600cc, 750cc, 1000cc, or 1200cc. In the last 6 years or so most have gone to the 1000cc size.
You simply buy a late modlel motorcycle engine and bolt it in and you are ready to go. Much cheeper than doing all the work required for a TQ motor. There are some Mini sprints out there for rent to see if that is really what floats your boat befor you invest.
I would be glad to talk to you any time about mini-sprints or as I prefer (1000cc midgets) My day phone is 502-778-6833 ask for Dad.
Honest Dad himself

