Definition of "intensive" and "extensive" properties

Today I was asked what does it mean for a physical property of a system to be intensive.

My first answer, loosely speaking, was:

“It is a property that is local.”

I was specifically thinking about density and, by “local”, I meant “that is unaffected by the dimension of the system”.
Ofcourse this is a very ambiguous answer, so after that I said (shifting to extensivity’s definition):

“A property is extensive if it depends on the volume of the system observed.“

To be honest, I said if it’s proportional to the volume, but I’m not sure the this is correct. Now, that I’m still thinking about it, I’ve come to the conclusion that a good definition could be:

“A property is extensive if it depends on the quantity of matter of the system observed.“

Looking on wikipedia I realized that this is exactly the definition given. But I’m somewhat still uncomfortable with that: if a gas is kept in a recipient of volume $V$ at a temperature $T$, his pressure is function of the number of moles of the gas:$$p=n(RT/V).$$
And, as we know, pressure is an intensive property. So (to me) it is not really clear what does “does not depend on the quantity of matter” mean.

I also thought that one could use an operational definition (if this is the good term) of extensivity/intensivity: one example might be:

“Suppose to measure a quantity $q(S)$ relative to a system $S$. Now reproduce a copy of $S$ and measure the same quantity for the system $S+S$ given by the two identicaly systems joined. If $q(S+S)=q(S)$, then $q$ is an intensive quantity.”

This seems to give a more precise sense to the “does not depend on the quantity of matter” in the above definition, but there are gaps to fill. Maybe I will try to develop better this in a second time. Now, ofcourse, the question is: what is the definition of extensivity/intensivity in rigorous and unambiguous terms?