Yes, Peter, I think that is the way it was probably done - although I am not expert enough to be 100% certain. I guess I would want you to look at the vase as if it had no engraving or signature - where might it be from. I suspect China, Poland, or one of the other generic makers.
You should realise that many glass firms(some quite significant) did not make their own glass or crystal. They bought in blanks and did their own cutting or engraving. For instance there are quite a number of Irish firms that produced small cut crystal items. It is my understanding that only Waterford actually produced the crystal they used. Similarly many smaller Swedish makers bought blanks from Orrefors or Kosta/Boda and made a business out of engraving them for sale.