I'd call it a slightly modified A-section, not much different than the 1st/2nd endings you often see. If it would make you more comfortable, I suppose you could label it C-section. I wouldn’t.
Rehearsal marks simply make it easier for the musician to organize the tune at first look.
It identifies the structure of the form. AABA, AB, AABC etc.
In "rehearsal" it makes it easier to identify a specific measure under discussion. (eg. B, bar 3) In practice, folks might refer to the sections as A1,A2 and A3.
Rehearsal marks are supposed to make things
less confusing.
In iRp, the A mark also identifies the start of the form for the iRp player. When the player repeats the form, it will return and restart there (instead of the verse or intro at the top of the chart).
As to the double bar to enclose sections, that conforms to our chart layout conventions
https://forums.irealpro.com/threads/iReal-b-layout-conventions&p=301.194/#post-301
Thanks for sharing your charts

)BOB