John,
I have built a rotary converter from scratch, I won't say how on here but it was basically the same design as my commercial unit.
They are not particularly hard to put together if you have the right bits and know how to set the voltages with the capacitors.
The real problem I had was getting a high enough voltage step-up transformer with enough current capacity. I used a transformer that is very common in the UK but it only gave me about 350V, a little shy of the 415V that was really needed.
It worked fine and would probably run my mill OK. But my lathe has a two speed motor and it just couldn't quite get the high speed going. If you spun the spindle by hand it would sometimes run up to speed. Sometimes not.
So I searched for a transformer winder and got some quotes. The price was frightening. Put it this way, if I added £100 to the price of the transformer I could buy a brand new commercial rotary converter, so that's what I did. It was worth it for the nice box it came in plus all the safety relays etc.
Re slip, it's not a constant, it varies with load as far as I know.
The two types of motor are synchronous and asynchronous, it's the latter used in the record players that use a disk/magnet to set speed.
I don't know which one our machine motors are, I forget so much over time.....

You can run a 3ph motor on single phase without a converter, but it will be reduced to 2/3 power. A friend runs his Harrison mill and lathe this way and reports no problems at all.
Anyone reading that please note you can't just put a 13amp plug on a 3ph motor, it just won't work and it's a little more complicated. But not much.
CC,
Sorry, it's a bit difficult to simplify the language. The terms are (as far as I remember) correct, if we used anything else John and I (or anyone else) wouldn't have a clue what the other was on about.............sounds like a normal course of events....!!!