Some engines like Briggs mostly and have seen others have a loop or more like a bent metal bracket with a hole or maybe two in them bolted to head bolts. They are not really very heavy,but figure that is what factory used to drop them into the tractor chassis when assembling. Maybe that is part of Specs and are ordered that way by assemblers yet don't come on replacement or new engines sold seperate? I would think you could make your own bracket and use headbolts OR, as you noted an eyebolt for temp use. If you make a bracket, you could leave it on engine all the time. Eyebolt would have to come off each time. Either way should re-torque those bolts loosened or maybe all of them when done.
I have used small lift slings, cloth and looped onto driveshaft, around the blower housing and other items with my engine lift. These are a small loop of cloth strapping made for lifting. Sometimes you need more than one to get like three areas lifted on so as not to just tip whole thing over once in air. The straps are not as secure and you have to make sure not to lift on carbs or other soft items that might get damaged.