Carbocation rearrangement with expansion of five-membered ring?
Your proposed mechanism involves a ring expansion from cyclopentane to cyclohexane. This gives very little driving force in terms of relieving strain in the system. Cyclopentane, in a twisted or "envelope" conformation, has some angle strain ($102-106^o$ vs $109.5^o$ $^{ref~1}$) and it does also has some torsional strain down some of its bonds. 

This is mostly alleviated by the move to cyclohexane, if it is in a full chair conformation, but the stabilization of the system is minimal ($25-29kJmol^{-1} $ $^{ref~2}$). Here is a link to a question that deals with ring strain and the calculation of it from $\Delta H$ of combustion.

Along with this, your mechanism proposes this ring opening involving an alkyl shift that moves the carbocation from tertiary to secondary. This involves a destabilizing of the cation due to the reduced hyperconjugation (and inductive effect). Studies have shown that a secondary carbocation is around $67-75kJmol^{-1} $less stable than a tertiary, for simple substituents (and in the gas phase$^{ref~3}$).

In total, the ring opening from a five to six-membered ring in your mechanism does not result in an overall increase in stability. Hence why the second mechanism is suggested to happen instead.