I have done a thorough search of literature relevant to Einstein's published attempts to unify electromagnetism and gravitation. Although Einstein's original papers (1),(2) on the subject do not seem to be available in English, there are two later papers by Einstein (3),(4) that summarize that work. There is also a good paper by Hubert F. M. Goenne (5) that, by reference to Einstein's correspondence, reveals his reasoning and efforts with respect to unifying electromagnetism and gravitation. There is probably a lot to be learned by studying Einstein's correspondence with other theoreticians, but I haven't had time yet to do that.
The bottom line, however, is that Einstein rejected any approaches that did not result in new physics such as explaining the existence and properties of electrons as solutions of the unified field equations. For example, the “Einstein-Maxwell” equations would not be considered a unification of E&M and gravitation because they simply provide a way to clump Maxwell's equations together with General Relativity without suggesting any underlying necessity for doing so other than convenience. And, of course, he rejected approaches whose results conflicted with known experimental results. All approaches that he considered failed one or the other of those two tests.
- Einheitliche Feldtheorie von Gravitation und Elektrizität [http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/echo/einstein/sitzungsberichte/ZHDAXTXQ/index.meta]
- Neue Moglichkeit Fur Eine Einheitliche Feldtheorie Von Gravitation Und Elektrizitat [http://alberteinstein.info/vufind1/Record/EAR000078395]
- A. Einstein, A generalization of the relativistic theory of gravitation, Annals of Mathematics Vol. 46 No. 4 October 1945 [http://www.jstor.org/stable/1969197]
- A. Einstein and E. J. Strauss, A generalization of the relativistic theory of gravitation (II), Annals of Mathematics, Second Series, Vol. 47, No. 4 (Oct., 1946), pp. 731-741 [http://www.jstor.org/stable/1969231]
- On the History of Unified Field Theories, Living Rev. Relativity, 7, (2004), 2 [https://link.springer.com/article/10.12942/lrr-2004-2]