I think that's the correct answer, but I think your reasoning is flawed.
You have to think of the mechanism in steps. After each step the species that are present change and that changes what the likely next step is.
So to go through an SN1 mechanism step by step, using this problem as an example:
1) You have a strong acid (HCl, which is likely dissociated into H+ and Cl- ions) and a somewhat basic substrate (the alcohol). When you have a strong acid/base and a weaker base/acid, it's usually a good starting point to bring these two together. So, protonate the alcohol.
2) Now there is a protonated alcohol and a Cl- ion in solution. The alcohol in this case is at a very hindered position (tertiary carbon) so SN2 is very unlikely. You do, however, have an exceptionally good leaving group (water), which can leave spontaneously. So have the water leave.
3) Now there are three species in solution: water, Cl-, and the carbocation from where the water left. This carbocation is quite stable since it is tertiary, so it can linger in solution for a while, but the positive charge still wants to be quenched. At this point there are two things that could happen in order to quench the positive charge: elimination or SN1 substitution. Since Cl- isn't a very strong base, elimination isn't likely. Cl- is, however, a pretty decent nucleophile, so it can attack the carbocation and form 1-chloro-1-methylcyclopentane.
So things are more subtle than just whether or not the nucleophile is charged.