Why do different manufacturers recommend different throttle settings at startup?
It's just a ballpark value that someone decided on during testing, that provides enough air to the engine to let it start and run with a rich primed mixture, without causing too much initial RPM.  A quarter inch here, and eighth inch there, it's all good as long as it doesn't cause the engine to burst into life at high power, and nobody expects you to sit with a measuring stick to get it exactly right.
It's not a good idea to start an engine with a very high throttle setting, because high rpm before there is sufficient oil flow = bad in the long run.  Crank and rod bearings need to be fully pressurized before any significant load is put on them. Cam followers need to be well coated in oil before they spin at more than minimum speed (especially with Lycomings).
Most engine wear happens in the first 15 seconds of starting, when the oil flow is inadequate and bearings, piston rings, cam lobes and gears etc are running on only a residual oil film.  Once a plain bearing is fully pressurized with oil, there is no metal to metal contact and the life of the bearing, in theory, is infinite.  If you ran a 2000 hour TBO engine with just one start and let it go without ever shutting down and just kept it supplied with fuel and oil 24/7, it would run far far beyond its normal TBO (although it might need cylinders at some point as valves wear out).
My personal practice is the use the absolute minimum throttle opening that will let the engine start, and let the engine tick over at minimum speed until I see normal oil pressure, before I go to a warm-up RPM. When an engine is warm, that usually allows starting with the throttle completely closed. Oil gets to where it needs to pretty fast; it's just that initial 10-15 seconds that's critical.