There are absolutely diamond fashion rings out there that are not marketed as engagement or promise rings.
A ring is an engagement ring if it signifies an engagement. Even if you are getting engaged and the ring you purchase/your partner purchases is not labeled as an "engagement ring" or doesn't look like a "traditional engagement ring," it's still an engagement ring if it is worn to signify an engagement. Couples can also get engaged without a ring at all.
And even if you go out and buy something labeled as an "engagement ring" for yourself, it's not actually an "engagement ring" if you're not engaged. That being said, if you wear a diamond ring on your left ring finger then people might jump to conclusions and assume it means you are engaged ... that doesn't make them right and it doesn't make you wrong, but just be prepared for some questions or for people to assume you're engaged and maybe avoid asking you out. It's fine to respond with, "I'm not engaged, I just like this ring." Nothing wrong with that. Just be aware that the "traditional" meaning is to send a message about your impending marital status.