Regarding (2), some evidence that Baldwin refers to some sort of $L_{\omega_1 \omega}$ version of Morley's theorem, rather than just an alternate proof making use of $L_{\omega_1 \omega}$ machinery, comes from a 1970 survey by Keisler himself. He mentions that "various forms" of Morley's theorem were extended to $L_{\omega_1 \omega}$ by "Choodnovsky [sic], Keisler, and Shelah, 1969" (p.149) though no citation is included in the references. And a look through the Shelah archive seems to turn up no relevant joint work with either of the other two.
I don't have a copy on hand, but one promising source for clarification (beyond inquiring with Baldwin about the content of his slides) is Keisler's 1971 book Model Theory for Infinitary Logic, which likely covers the result(s) in question such as they are; and though perhaps only a coincidence, that does match the year Baldwin's slides assign to the matter.
ETA: Baldwin's Categoricity book confirms both the nature of the result and his direct source: "Keisler [Kei71] generalized Morley’s categoricity theorem to sentences in $L_{\omega_1 \omega}$, assuming that the categoricity model was $\aleph_1$-homogeneous" (p.22). Though Baldwin points to Keisler's book as the basis for transferring Morely's theorem to infintary logic, he also attributes most of the machinery to Shelah (p.xi).