How can I get the size of an std::vector as an int?

In the first two cases, you simply forgot to actually call the member function (!, it’s not a value) std::vector::size like this:
#include

int main () {
std::vector v;
auto size = v.size();
}

Your third call
int size = v.size();

triggers a warning, as not every return value of that function (usually a 64 bit unsigned int) can be represented as a 32 bit signed int.
int size = static_cast(v.size());

would always compile cleanly and also explicitly states that your conversion from std::vector::size_type to int was intended.
Note that if the size of the vector is greater than the biggest number an int can represent, size will contain an implementation defined (de facto garbage) value.