Ahhh, good old crystallography causing confusion. Relax, it is not uncommon to say the least. Only long, deep study of point and space groups will lead to clarity.
However, there are good resources available on line, including the University of London. There you will find the following quote:
Each one of the 230 three-dimensional space groups is unique; however, they are frequently specified by means of space group symbols that are not unique to a particular space group. The reason for this is very simple: while each space group symmetry is unique, the choice of vectors that defines a unit cell for that symmetry is not unique. It was mentioned earlier, for example, that the unit cell for the centred monoclinic Bravais lattice is conventionally described as C-face centred with the unique symmetry axis direction being parallel to b. Space group symbols all begin with the lattice type as the first character of the symbol, e.g. C2. With a different choice of labelling for the unit-cell axes, the same space group (number 5) could easily be called A2; other possible symbols are I2, B2, or even F2. You should therefore be aware that the symbols used in the table below refer to so-called standard settings (and hence standard symbols) for the space groups.
So, yes, by choosing a different basis, you describe it differently. Pick the one you prefer or are most comfortable with (likely C2/c) and go from there (and C2 would be more common).
After the wonderful comment by @marcin (that might disappear) I think it really should get into this answer. @marcin pointed out a really nice paper (J. Res. Natl. Inst. Stand. Technol. 107, 373-377 (2002)) which discusses standardization of some of these multiply-definable crystal structures. Here the author (Alan D. Mighell) argues that the centered monoclinic lattices, which one-third of the literature references with the C convention, should instead be more clearly listed under the I convention. After reading the article, I think it has much merit, particularly if you deal with monoclinic systems often (I don't). So, in my answer above I was incorrect in suggesting using the C2 convention, and that it was more common. I was letting my rectilinear biases come into play (my simple mind prefers cubes I guess).