Counting basis sets in quantum chemistry I have to use STO-nG (a "minimal basis set" - meaning that only one basis function is used for each atomic orbital in the atoms of which the molecule is made from). Let's take the example of a water molecule.

The water molecule has two H atoms and one O atom. Thus, we have a total of 7 orbitals (two 1s of H, one 1s of O, one 2s of O and three 2p of O). So when using STO-nG, would that then mean that three 1s type basis functions, one 2s type basis function, and 3 p-type basis functions are used, totalling at 7 basis functions, each being made up from a linear combination (LC) of n simple gaussians?
Yes, you have counted the number of basis functions correctly:

    #1s    #2s    #2px    #2py    #2pz    #Total
H1   1                                      1
H2   1                                      1
O    1      1       1       1       1       5


Which as you also counted is 7 basis functions.
As you also correctly state, each of the basis functions will be a linear combination of $n$ primitive Gaussians:

$$\phi(r)=\sum_{i=1}^nc_i\exp\left(-\alpha_i\left(r-r_0\right)^2\right)$$

As you have observed that is what the $n$ in STO-$n$G denotes.