Q: What is the history of Eisenstein series? Did the mathematician Eisenstein actually encounter them?What is the history of Eisenstein series? Did the mathematician Eisenstein actually encounter them?
A: Yes, he did, see Elliptic Functions According to Eisenstein and Kronecker.
Eisenstein writes the double sum (on page 223) as
where $w=\alpha m+\beta n$ and the sum is over the two integers $m,n$.