With the level-plug Briggs you are supposed to park it on level ground and fill with suitable (SAE30, not multigrade) oil to the top of the thread that retains the level plug - in other words, fill until you can't put any more in. For some reason my 92908 does not cooperate with that approach: if I fill to that level and leave it parked in the shed (level concrete) it expels oil through the plug-to-crankcase joint onto the floor. If I fill to the bottom of the thread I have no problems, so that is what I do.

Your 93508 will have a Vacu-Jet carburetor. I don't know if that fourth design generation still had the automatic choke. The Vacu-Jet was a cheaper carburetor than the Pulsa-Jet, with no fuel pump. It just relied on venturi suction to pull fuel up from the fuel tank through the jet to the venturi. As a result, air fuel ratio depended on fuel level in the tank, and they tended to be slightly harder to start (especially with low fuel level). Nevertheless my 92508 is a nice engine, starts quite easily, and once running is quite satisfactory. Your fuel-gauge-in-the-cap sounds like a fourth generation feature - I don't know what else was different.

Identifying a small vertical Briggs without cowl or carburetor sounds difficult, but there are Outdoorking members who know way more about this than I do. If you post a picture of the engine, there is a good chance someone will be able to help.

Your Masport/92908 project sounds like one that will interest members. Please post pictures and reports. I'll leave this thread open.