The distinction is a bit more subtle.
The "large G" symbol isn't actually a rune, it's handwriting; to be more specific it's the letter "G" in Tolkien's Feanorean alphabet (Tengwar).
The "Elf-rune" "G" is however a rune, and it's also of Elvish origin. This is given in Appendix E(ii) to Return of the King (the Angerthas tables) where it's clear that it's origin is the runes of Doriath (Cirth) devised by Daeron.
In the Dwarvish runes used in the Hobbit, this symbol was actually used for "F" and the rune for "G" was different.
So there was no switch; both letters given are Elvish, but only the second is a rune.