Basically PATN-1 and 2 (and whether they carry both patn copies like PATN-1PATN-1 or not) just influence the "look" of the appy (ie: whether it shows up as fewspot, snowflake/blanket/frost, leopard, etc.) One thing to note is that you can have a PATN-2 horse and it just has "appaloosa" without a blanket/frost addition, but this just requires a reroll to get one of the PATN-2s to show up (they just sometimes juke you and look plain appy xD). But if the horse just has Lp with neither of the PATNs, it will always remain plain "appaloosa" with no visible spots except for muzzle molting.

Below is copied from the game guide, but I couldn't say it any better xD I can always explain further through PM if you need!

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In this game Lp determines if your horse is Appy, and PATN determines what kind of pattern they have. Appy genes work as follows:

• Lplp patn-1patn-1 patn-2patn-2 Appy with no appy markings
• LpLp patn-1patn-1 patn-2patn-2 Appy with no appy markings
• Lplp PATN-1 Leopard
• LpLp PATN-1 Fewspot or Snowcap
• Lplp patn-1patn-1 PATN-2 One of the following: Blanket with loud spots, Snowflake with less white, Varnish with less white, Frost with less white
• LpLp patn-1patn-1 PATN-2 One of the following: Blanket with small spots, Snowflake with more white, Varnish with more white, Frost with more white

Varnish can appear in conjunction with other appy patterns, however it only appears by itself in Lp? patn-1patn-1 PATN-2? horses.