There are no rules or requirements for an engagement ring, or a wedding ring. Rings aren't required to get engaged or married in the first place.
Any ring you may choose to give for an engagement is then an "engagement ring," whether it's marketed as such or not. It doesn't matter where you got it, what it's originally called, what it's made of, what it looks like, how much it costs, what metal/gemstone is used, etc.
So, yes, if you use the ring that you like in order to signify an engagement, then it is indeed an engagement ring.