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After looking more into it, the pokemon that will be sent out first when a battle starts will the one from either player who selected it first.
I always take time choosing my lead in tournament matches and when my opponent has already did, the lead animation is coming from them.
Whereas in...
What type of bug are you reporting? Unsure
What is the bug?
Challenging an account of an ongoing Ladder Tournament will force the battle to be public even though it is not a ladder battle.
https://replay.pokemonshowdown.com/gen9nfe-2044832905...
Responses on that from Marty way back:
https://www.smogon.com/forums/threads/gen-8-smogon-university-usage-statistics-discussion-thread.3657197/post-9352655
https://www.smogon.com/forums/threads/gen-9-smogon-university-usage-statistics-discussion-thread.3711767/post-9564335