Deuring's result on elliptic curves. Any proof reference

You might find the following paper useful, although it proves something more general than what you are asking:

MR0890272, Rück, Hans-Georg, A note on elliptic curves over finite fields. Math. Comp. 49 (1987), no. 179, 301–304, doi:10.1090/S0025-5718-1987-0890272-3.

There is also the paper:

MR0265369, Waterhouse, William C., Abelian varieties over finite fields. Ann. Sci. École Norm. Sup. (4) 2 1969 521–560, doi:10.24033/asens.1183.

The review of this second paper says: "In Chapter 4, Deuring's results on elliptic curves are derived, where the classification is very explicit."