Differences among Doctoral Degrees/titles
An EngD is a different degree to a PhD. If you've got a PhD you can't put EngD after your name, and vice-versa. Both can give you the title "doctor", but they are different doctorates.

If your degree diploma says "Doctor of Engineering", it's very very likely that you can put Eng.D. after your name. But not PhD. (and I've never seen D.Eng after a name, only Eng.D. - but maybe that's UK-specific. I have seen Dr.-Ing. from Germany)