Interesting stuff - I found this on Wikipedia after reading your post:
"The team was officially named the "Senators" during
1901–
1904, the
Nationals during
1905–
1955 and the Senators again during
1956–
1960, but nonetheless was commonly referred to as the Senators throughout its history (and unofficially as the "Grifs" during
Clark Griffith's tenure as manager during
1912–
1920).
[1][2] The name "Nationals" appeared on the uniforms for only two seasons, and then was replaced with the "W" logo. However, the names "Senators," "Nationals" and shorter "Nats" were used interchangeably by fans and media throughout the team's history; in
2005, the latter two names were revived for
the current National League franchise that had previously played in
Montreal."
I thought they were the Senators the entire time from 1901-1960, but I guess I was wrong...