Types of Assassins?

I've been pondering assassins lately (for the obvious reasons -- assassin update) and I found it striking that champions like Rengar, Katarina, and Shaco could all be considered in the same boat here. (Don't worry, I've done my research here: http://na.leagueoflegends.com/en/news/game-updates/gameplay/dev-blog-classes-subclasses and here: http://boards.na.leagueoflegends.com/en/c/developer-corner/3A5uuBw7-champion-subclass-list )

And so I've come to the conclusion that while they all fall under the same title, there are actually several massively different types of assassins. Now, help me here, but so far I have two good types, and I'm kind of grasping for a third.

Burst Assassins: Assassins such as Rengar, Zed, etc. that have the potential to deal massive bursts of damage in a short space on a single target, and whose play-styles generally streamline them into this option. (Note, I consider Xin Zhao to potentially fall in this category, but he's not even technically an assassin.) These are sort-of the "Classic Assassin" type.

Stealth Assassins: Assassins such as SHACO, ~Rengar, ~Le'Blanc, and Kha'zix (and, I argue, Evelynn--again, not technically an assassin) that use stealth to sneak up on their targets and scare the #### out of them, or kill them flat-out depending. While definitely capable of burst, these champions rely on the element of surprise rather than just damage.

~? Assassins: Assassins such as Talon, Fizz, and Kassadin that use mobility, positioning, and utility (stealth, invulnerable, CC) to outplay their opponents and leave them with no other option but death. I'm not sure what I'm grasping at with this, but I feel like these champs have kits that combine burst with dueling power and utility that pushes them towards a certain play-style they all kind-of share.

Anyway, it seemed strange to me that all of these diverse champions were grouped under such a loose description, and I don't see how Riot can plan to update them with such a limited scope as of now; the current definition is essentially: "mobility and burst." Which encompasses a rather broad (and often annoying) spectrum.

Please, give me feedback here. I'm curious. Oh! And have a good day.

~CrimsonCobra3