Wrong, hot wallet isn't limited to online wallet which controlled by third-party. Hot wallet refers to wallet which connected to the Internet.
This. You're describing a custodial wallet with that description OP, not necessarily a hot wallet. You can have a hot wallet that's not controlled by a third party (e.g. Mycelium on Android, etc).
P.S. i don't see any points of this thread
*drumroll please!*
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🎺🎺 Merits!