Alkene reaction with acid via SN1 reaction

I want to ask a question about alkenes and their reactivity with acids.

From high school studies, I recall that if you had an acid $$\ce{H3O+}$$ i.e. $$H^+$$ that it would react with an alkene as shown below:

However, when I was given a lecture today on Allylic alcohols in a $$\ce{S_{N}1}$$ reaction, the acid does not react with the alkene at all:

As you may understand, attempting to google or find an equivalent mechanism to explain this observation didn't yield a successful result.

The two mechanisms in my opinion appear to contradict each other, but I couldn't work out how the second mechanism for the Allylic Alcohol functioned whereby the alkene did not react with the acid. My assumption was that it was easier to protonate the alcohol group than react with the acid (with maybe a reasoning along the lines of pKa) but I couldn't find anything sufficient to justify my idea.

Why does the acid not react with the alkene group in the allylic alcohol shown above?