Can a ladybug be determined just by its number of points?
No, most of the time you cannot. There is a large diversity of ladybug species (several dozens in western Europe), and many of them can have the same number of spots. Moreover, as you already said, there can be intraspecific variation in the number of spots.

I do not know if there are specific numbers of spots that would be associated to only one species - but for most numbers, it is not the case. For instance, yes, it is common for Harmonia axyrids to have 18 spots (source). But Anisosticta novemdecimpunctata, which  can also be found in Europe - and can also be orange ! - may have between 15 and 21 spots (though it more frequently has 19) (source).

Thus, the number of spots cannot be used as a sole criterion to identify ladybugs: other criteria usually include the general shape, colour of elytra, size, shape and colour of the pronotum, length of antennae, etc. (See here for examples.)

First picture: Harmonia axyridis



Second picture: Anisosticta novemdecimpunctata