Rearrangement in dehydration of alcohol
Concentrated sulphuric acid is a highly acidic medium and thus it very exclusively prefers the $E_1$ mechanism over $E_2$ mechanism and thus a carbocation is surely formed in this conditions.
The example you provide proceeds through the following mechanism:

Now the mechanism you provide, other than following the wrong pathway(very minor pathway) also has a major mistake. During the protonation of 1-methylenecyclopentane the tertiary carbocation is not more stable than the primary carbocation. The tertiary carbocation is $\mathrm{sp^2}$ hybridized and thus prefers a bond angle of 120° which cannot be provided by the cyclopentane ring as it is not flexible enough.
It prefers the very less angle strain in case of $\mathrm{sp^3}$ hybridized orbitlas but with less positive inductive effect over higher angle strain but more positive inductive effect. That's the reason the below equilibrium is highly backward favoured.