Yes, it would seem logical that the t213-2 versions of Germany were issued as Washington 1914, Newark 1915 and New York 1916.
According to Lew Lipset's book, the t213-3s that contained no team designations were retired players, coaches, etc. Schaeffer (sic) was no longer active in 1919, so this would explain why that card has no team designation. It also suggests re-use of something other than the 1916 card, since that issue had no W emblem on the uniform.
The 1915 t213-2 is really interesting. Schaefer did not sign with anyone until around the 7th of March, so it seems the cards would not have been issued by then. In fact, there was some question about the Newark team itself, which wasn't even named until about that time and which still had its park under construction (it was a transplant for the Indianapolis Feds from 1914). A March 6 article in the Sporting Life referred to reports from Kansas City that doubted whether Newark would have a team. One wonders if those same reports had Schaefer landing in their town, and if so, that is what prompted Coupon to caption him there. Anyway, Schaefer signed with Newark.
Now at the end of April, for about a week or so, there was rumor and even a report of a multi-team trade that would have sent Germany Schaefer from Newark to Kansas City. That deal fell through, but it makes it curious as to whether that is when the t213-2s might have been printed. That is another possible explanation. From 4/24/1915:
from 5/1/1915:
In any event, it does not appear that placing Schaefer on the KC Packers was a simple or random mistake.