I've seen two meaning of 'with high probability' in the literature, and I think they are different. I am not sure whether I miss understood them. Here are they:
(1) In mathematics, an event that occurs with high probability (often shortened to w.h.p. or WHP) is one whose probability depends on a certain number n and goes to 1 as n goes to infinity, i.e. the probability of the event occurring can be made as close to 1 as desired by making n big enough. (source: Wikipedia) For example here An event Π is said to occur with high probability if $P(Π)\geq 1-\frac{c}{n^\alpha}$.
(2) In the context of concentration, if we have $P(X\geq a+t)\leq \exp(-t^2)$, then we say $X<a$ with high probability. This means that the larger the deviation from what we expected ($X<a$), the smaller the probability is. For example, theorem 3.1.1. proved $P\{\|X\|_2-\sqrt{n}\geq t\}\leq 2\exp(-\frac{ct^2}{K4})$, and call it 'with high probability $X$ takes values very close to the sphere of radius $\sqrt{n}$'. Also, I noticed that in some concentration bound, it can depend on $n$ as well.
From my understanding, the first one is saying the probability is bounded by a function of $n$, while the second one is bounded by a function of deviation, which is commonly addressed topic in the context of concentration.
I am not sure whether 'with high probability' is a vague word, and its definition depends on the context, or I misunderstood something.
Could someone help to clarify more? Thanks in advance for any comment, answer.