API rating and viscosity are most important. The API rating should be dictated by the equipment or engine manual. Viscosity is dictated by the operating temperature of the motor. I am no chemist or expert, but there is likely a difference between small engine oil formulated for air cooled equipment. This may be the addition or subtraction of different detergents and/or friction modifiers.
Call me a nut but I use Honda oil in Honda engines, Exmark Oil in my Exmark, Cub Cadet oil in my Cub Cadet and Ariens oil in my Ariens. I think you get the picture.
I am still in my 4 year warranty on my Exmark so I only use what the dealer sells me, genuine filter and Exmark 10W40 Synthetic.
My Cut Cadet walk behind string trimmer gets 10W30 which the manufacturer was gone to the lengths of placing a nice sticker right onto the side of the engine.
In my Ariens snow blower they only offer a 5W30 oil from Ariens that is specific to snow equipmen.
Lastly, my Honda push mower is using 10W30 but that is due to the fact that the Honda SAE30 is getting harder to find at Home Depot and other places around me. Otherwise this is what I would use.
In reality, I could probably use just two oils for all of my equipment but I am too particular. I could be wrong but when the engine manufacturer, in this case Honda, Briggs, LCT or whoever is engineering and proving out the reliability and performance of their engine, I would think that they would be using their own oil. Or the opposite could be said, that when Honda or Briggs contracts a refinery or chemical manufacturer to "blend" a specific grade of oil, they would then do their own quality and performance tests in their equipment to ensure its suitability in their products. Either way, sticking OEM oil and most importantly, following scheduled maintenance, is the safest way to go.
If you do not have the choice of OEM oil, find a small engine oil with the correct rating and viscosity and you should also do fine. And for all of this I am speaking for typical, homeowner use. Severe-duty applications might require other types or grades of oil; again, referring to the equipment/engine manufacturer.
I hope that my own "anal" tendencies and opinion might guide you. Good Luck!