You are on the right track about size regulations, however they differed over the years. Thus from the 1940 Postal Laws & Regulations volume:

To illustrate parcelpostguys' reply: The card (as mailed) is indeed oversized, thus requiring the letter rate. Here is a fairly similar 3-part blank application form for a member of the military (intended to be mailed with free frank, thus no need to differentiate the 2 or 3 cent rate), rather than a civilian, but the illustration of the entire form serves the same purpose here.
A small tab is retained by the applicant, the middle section serves to get it to the OPA Mailing Center (with stamp and postmark), and the third portion is torn off and retained by the OPA Mailing Center for their records.


I believe the central third accompanied the new 3rd ration book back to the applicant, with the handwritten address peeking out through a window envelope, which also explains the large number of these "middle portions" surviving.
Rate description. With San Francisco having carrier service, I would lean toward calling the 2 cent option as a "local" letter rate.
Here is a pair of cards addressed to Indianapolis showing both the local and the inter-city letter rates (with the Danville card being mailed slightly before the June 1-10 mailing period opened, but apparently without any penalty):
