BTW this room is now four years old. It seems it was rather active in 2015, 2016, 2017. (But not so much lately.)

BTW the name of the room changed a few times during those year.

The answer is fine. I don't understand your confusion.

Their point is that if you have two different points, then sufficiently small neighborhoods of these points don't intersect. Germs are equivalence classes of functions on open neighborhoods of your point.

When your neighborhoods don't intersect, the two functions on these neighborhoods have nothing to do with one another.

If you want to be really paranoid, argue that one can choose the value of the function at q independently of the value on a neighborhood of p. As Martin said earlier, this uses the "normal Hausdorff" property.

In any case: do not get hung up on this. This is not an important point, and you can immediately change your definition to avoid technicalities like this.