I genuinely don't even understand it. I was either taught wrong, learned wrong or this doesn't make sense. O(100) is constant time, so it's no different to saying O(1) or whatever, I've only ever used O to talk about how we care about the time complexity of a function, e.g. comparing linear time to quadratic time. 1 is the same as 100, irrelevant.

Agreed! I'm pretty sure they're just using O(x) to mean on the order of x, since big O of any constant is the same (unless that's their point??? :O)

If it’s big O notation, it makes no sense because O(10000)=O(1000)=O(100)=O(1).

I think they're trying to communicate that those are worst case latency numbers. You're correct in that formal use of Big O notation doesn't distinguish constants.

I would read ~100 to mean "approximately 100", while O(100) to me reads like "measured in the hundreds". So 200-600ms cold starts are O(100) and 3-7s cold starts are O(1000).

I'm not sure, I'm pretty sure they're trying to communicate that these are worst case estimates. ~100 seems like it's communicating average case estimates (big theta notation), which isn't really the same thing

Big Theta isn't "average case". It describes an algorithm with the same asymptotic lower and upper bounds. How would you even formally define an average case, when the result depends on the input?

It's that easy to raise money these days, at the 10% inflation mark. Spin up a webpage full of nonsense promises and they'll come chasing after you to take their cash.