Mechanism for the addition of hydrogen iodide to 3,3‐dimethylbut‐1‐yne
There seems to be an issue with both mechanisms on the front that an SN1 reaction would not take place since the carbocation formed is a vinylic carbocation which is highly unstable.
The actual reaction follows a termolecular mechanism where the rate of reaction is given to be:
$$\text{Rate}=[\ce{HX}]^2[\text{alkyne}]$$
Now, according to Advanced Organic Chemistry by Francis A. Carey, the reaction mechanism would be as follows:$^1$

Step $1$:
A concerted termolecular reaction...
This involves an acid/base reaction, protonation of the alkyne developing positive charge on the more substituted carbon. The π electrons act pairs as a Lewis base.
The other part is attack of the nucleophilic bromide ion on the more electrophilic carbocation creates the alkenyl bromide.

Step $2$:
In the presence of excess reagent, a second protonation occurs to generate the more stable carbocation.$^2$

Step $3$:
Attack of the nucleophilic bromide ion on the electrophilic carbocation creates the geminal dibromide.



[$1$]: The reaction mechanism stated above uses $\ce{HBr}$ and not $\ce{HI}$ instead. However this reaction takes place for $\ce{HX}$.
[$2$]: The same reaction using $\ce{HI}$ would have a comparatively lower yield of the geminal product since geminal diiodides are unstable due to the steric hinderance posed by the large size of the iodine atoms.