The best reference for such questions is Brills’s New Pauly.
Eckart Olshausen (Universität Stuttgart), the author of the book Einführung in die Historische Geographie der Alten Welt (1991), writes that “[a]s a term of Roman imperialism Mare Nostrum first occurs in Caes. B Gall. 5,1,2, whereas mare mediterraneum is a medieval word formation (cf. Isid. Orig. 13,6,1)” (Olshausen, 2006; emphasis mine - Alex B.). Mare internum seems to be the most common choice in the Historical Atlas of the Ancient World.
Olshausen also mentions, besides the above-mentioned mare nostrum, some other possible variants, mare internum and mare intestinum (Plin. HN 2,173 and Flor. 2,13,293).
The Barrington Atlas of the Greek and Roman World similarly adopts “Internum Mare.”