Was there any official distinction between the IBM 6580 Displaywriter System's beam spring and Model F keyboard variants?

Yes! The beam spring keyboard is known as Type A and the Model F keyboard is known as Type B. The February 1983 revision of the IBM Displaywriter System Product Support Manual describes these terms on Chapter 5 (Keyboard), specifically on page 64 of the bitsavers' PDF scan of this document. Figures showing the Type A and Type B assemblies are visible on page 63 (Figure 5-1) and page 64 (Figure 5-2) respectively.

What do the 'a with caret/chevron' and 'a/aaa with cross-out' keys on Model F104, F122 and early M122 keyboards do?

For the keyboards' original terminal layout, the 'a with caret/chevron' key functions as an insert key and 'a/aaa with cross-out' functions as a delete key. Evidence for this can be found by examining the Model F-based IBM 3270 PC Keyboard Element (such as P/N 6110344) - specifically, by comparing the physical layout (credit to John Elliott) with a known functional layout diagram (credit to John J. G. Savard).