I wasn't assuming how anybody things. I was literally reciting the foundations of your argument.
Teemo sits in wait.
This could literally apply to ANY champion.
His entire kit plays off assassin-style surprise, and thus to me, he is an assassin.
That's ridiculous. It's not based on "surprising" the target, it's based on the kit and playstyle.
Usually if he engages, he wins the encounter, like any good Teemo would.
Not necessarily. I've killed Teemos who had caught me off guard.
Zed's don't go in to poke, they go in to 100-0 in a combo.
Yes, because he's melee. If he was ranged, he'd poke from away, and THEN go in. Instead, he uses his combo to poke, and THEN go in.
Teemo can do that - especially if stealthed - but it is tick damage that is nearly impossible to mitigate.
None of the assassins rely on tick dmg like Teemo does, with one exception.

Talon's Bleed is useful for kills early, but all of his kills come from the pure burst mid-game.
The thing with Teemo is that he relies on his tick dmg. Zed doesn't. Talon doesn't. Leblanc doesn't. Kha doesn't. These are all actual assassins/assassin mages/assassin divers.
Plus invisible shrooms just when you thought you got away.
That could apply to anyone. The difference between Zed's ultimate pop and stepping on a Shroom is that the Pop isn't necessary. It just amplifies the damage, and is useful if you managed to deal enough damage to back out. Shrooms provide both defense and offense. If you're running away from a Rengar as Teemo, you pop a Shroom down behind you to escape. if you're hunting a Rengar as Teemo, you pop a Shroom down in a place, and bait him to that area so that he steps on it when he's running away. A Shroom does not add to any classification of an assassin. Shaco works the same way, with the exception being that Shaco also has proper mobility and damage.
Oh no, you're running from a Cait, but O shit there's a trap! ur ded.
Oh no you're running from a Nid but o shit there's a trap! ur ded.
Oh no you're running from a Zoe but o shit there's an E puddle! ur ded.
Having one extra trap as insurance in case you don't get the kill with your initial attack doesn't help classify one as an assassin. If Teemo were to be a real assassin with a kit-lore connection, his kit would be something not too different from Shaco's, with a few changes here and there. His stealth and shrooms would be fine. But his speed steroid would be replaced with a short range grapple. which can launch him over terrain. His Poison could be used in an Auto-Empowerment (think Kass W). His Q could be a skillshot slow which marks enemies. This'd give him both the qualities of an assassin, as well as a higher skill level, because face it. You literally just spam right-click and Q to kill people currently. With this, you'd have to use a variety of abilities to out-maneuver your opponent.