The intuitive notion of a category is given by the dependently-sorted algebraic theory of categories.

In the ZFC metatheory, a standard model of this theory yields the usual mathematical concept of a category.

In the MLTT metatheory, a standard model of this theory yields the concept of a precategory.

The subtle issue is how to treat the equality predicate, appearing in the identity and associativity axioms.

In set theory, equality is a former of atomic formulae, which are a different syntactic category than sets (which interpret the types/domains of the given language).

In type theory, equality is interpreted as any other type, in accordance with "the formulae-as-types notion of construction".

However, more modern type theories, like HoTT or CubicalTT, have a different, "native" conception of equality, so the "natural" notion of category in these theories will accordingly be different as well.

One day, there might be a metatheory which is generally accepted as providing a superior "natural" notion of a category, that avoids the awkward points of the set-theoretic one.

Until then, it is best to be precise with language, always being explicit which specific concept is meant.

Cheers,

Andrew