organic chemistry - How to explain the selectivity between methanol and methanethiol in an SN1 reaction with an halogenated hydrocarbon?

Consider the S_N1 reaction shown below:

Would different products be formed if $\ce{CH3SH}$ was an additional reagent?

Superimpose both energy diagrams on top of each other.

For the first question, yes, two products would be formed. I'm confused about the second question though. I know The rate determining step would remain unchanged, as the rate depends on the concentration of the substrate not the nucleophile (S_N1 is unimolecular). However, methanethiol is a better nucleophile than methanol. Would the activation energy for the second transition state be less for methanethiol?