2012-2022 are identical except for colours and the extra borders in some. However, the 2013 one uses a 2-coloured flame instead of a monochrome one. There are also a few cases where the same flame was used outside of the logo.
For those maybe like this (only structure, not names)?
Prim 1 - (c16) Flame Left 2012 (used in Yellow, Orange, Br-L-Yellow, Red and a few more used on minifigs)
Prim 2 - (c16) Flame Right 2012 (used in Yellow, Br-L-Yellow, Red and a few more used on minifigs)
Prim 3 - (c16) Shield Inner 2012 (used in "unprinted", Silver, Red, Black)
Prim 4 - (c16) Shield Border 2012 Innermost (used in Red, Black, Br-L-Yellow)
Prim 5 - (c4?) Shield Border 2012 - 1 (only Red?)
Prim 6 - (c14?) Shield Border 2012 - 2 (only Yellow?)
Prim 7 - (c0?) Shield Border 2013 (only Black?)
The 2004 and 2007 ones seem to differ only in the outer borders as well. Though the flames are used without the shield as well.
For those maybe like this (only structure, not names)?
Prim 1 - (c16) Flame Left 2012 (used in Yellow, Orange, Br-L-Yellow, Red and a few more used on minifigs)
Prim 2 - (c16) Flame Right 2012 (used in Yellow, Br-L-Yellow, Red and a few more used on minifigs)
Prim 3 - (c16) Shield Inner 2012 (used in "unprinted", Silver, Red, Black)
Prim 4 - (c16) Shield Border 2012 Innermost (used in Red, Black, Br-L-Yellow)
Prim 5 - (c4?) Shield Border 2012 - 1 (only Red?)
Prim 6 - (c14?) Shield Border 2012 - 2 (only Yellow?)
Prim 7 - (c0?) Shield Border 2013 (only Black?)
The 2004 and 2007 ones seem to differ only in the outer borders as well. Though the flames are used without the shield as well.