You haven't gotten anything wrong, you are entirely correct!
A set of quantum numbers can be used to identify the specific electron in the sub-atomic world of an atom, and what path it occupies in three dimension.
$\ell = 2$ refers to the $d$ orbital. Now how many electrons can a d orbital contain? It has 5 "slots" because we can go from -2, -1, 0, 1, 2. These represent the orientations, each one holding up to two electrons at different spins (up or down, -1/2 or +1/2). We therefore know that $\ell = 2$ refers to a possible 10 different unique sets of quantum numbers, because each set could represent a specific electron, of which there could be up to 10 in the d orbitals, only 6 for p orbitals, etc.
The principal quantum number ($n$): Has only integral values (1, 2, 3, etc) starting at 1. Think of this as the power level. An electron in a 3s, 3p, 3d, etc orbital is in the third power level. It represents the numbers before the letter in the electron configuration such as 1s$^2$2s$^2$2p$^6$ so in other words, it represents what level the electron is on.
The angular momentum quantum number ($\ell$): goes only from 0 to $n-1$, also uses integral values like $n$. Think of this as the letter after the number in the electron configuration such has 1s$^2$2s$^2$2p$^6$ so in other words it represents which orbital we are talking about on that power level. 0 represents $s$, 1 for $p$, 2 for $d$, 3 for $f$ etc.
The magnetic quantum number ($m_\ell$): Goes from $-\ell$ to $\ell$ including 0. Think of this value as representing a particular orientation of an orbital. The number of orientations will vary depending on the orbital. s orbitals are spherical and can only have one orientation, s orbitals are referred to by saying $\ell = 0$. If we said $\ell = 1$ we would be talking about the p orbital, which has -1, 0, and 1 orientations (3 total). This can be thought of as the x, y, and z axis. $d$ orbitals have 5 possibilities, $f$ has 7, etc. I like to refer to each of these orientations as a "slot" which can hold two electrons.
The value for $m_s$ is used to identify the spin either up or down, but not both.