How do I determine if I have one of those "couple other phase converters" which really makes 3 phase? 

My 3 phase welder works fine as far as I can tell.  It will stick stuff together and it stays stuck together.  It is far more capable then the smaller single phase one, and has produced results on many jobs which I considered to be satisfactory. But then, I've been accused of not knowing anything about welding before so it's entirely possible everything I have welded with it just looks like it is built or rebuilt.