When I took my first class in biblical studies, I was a little surprised to find that scholars generally don't call the Old Testament the Old Testament. My Jewish professor (Saul Olyan for those who care) preferred not to bring in the connotations Christians associate with that term and simply called it the Hebrew Bible. I was fine with this for the sake of that class, though I preferred to use the standard Christian term in most contexts. I didn't like the term 'Greek Bible' for the New Testament, though, because no one thinks of the New Testament as a whole Bible. It's not the Christian Bible either, because that's both testaments.
I did know that some Christians didn't like the standard 'Old Testament' and 'New Testament' descriptors because of things they seemed to convey that might not be accurate. I didn't know that anyone had proposed replacing them with 'First Testament' and 'Second Testament'. If anything I would have preferred 'Old Covenant' and 'New Covenant', since 'testament' is generally a mistranslation (in contemporary English anyway) of the term for covenant in the New Testament. But I'm generally the sort to prefer the names we have already, because once something becomes a name it's really ceased to be a description at all, as evidenced by the countless inaccurate titles we use all the time that nonetheless succeed in referring to their intended designee (e.g. 'driveway' and 'parkway' are not descriptions but names of categories that seem to have reversed their etymological meaning, and 'the United States of America' no longer refers to a collection of nation-states but a bunch of provinces that we inaccurately call states).
Anyway, Tyler Williams has a great post on this: Old Testament/First Testament/Hebrew Bible/Tanak: What's in a Name? Quite a Bit Actually! He summarizes the different terms and the reasons offered for and against all of them in a way that I think is pretty fair to all parties.