I don't know much about large cardinals; so, I want to get a feeling of the landscape. Hence this question.
Definition. Whenever $C$ is a unary predicate in the language of ZFC, let us call $C$ semiabsolute iff for all transitive models $M$ of ZFC and all $x \in M,$ if $\mathcal{P}(x) \in M$, then $C(x)$ iff $C^M(x).$
Question. Do large cardinal properties tend to be semiabsolute? What are some examples of large cardinal properties that are not?