You can do the same thing for $HHT$ (and likewise for $TTH$) as you did for $HHH$. If I choose $HHT$, choose $THH$; unless the first two results are both $H$, with probability $\frac14$, $THH$ comes before $HHT$.
More generally, you want the longest prefix of mine to be a suffix of yours and the shortest possible suffix of mine to be a prefix of yours.
So if I choose $HTT$, you want $HT$ as a suffix and neither $TT$ nor $T$ as a prefix, so you choose $HHT$. If I choose $HTH$, you want $HT$ as a suffix and ideally neither $TH$ nor $H$ as a prefix; since you can't avoid both, you avoid the longer one, $TH$, and again choose $HHT$.
To calculate the winning probabilities in these cases, label the possible non-terminal states according to the two most recent results, and note that since both our patterns start with $H$, the initial state is equivalent to $TT$. From this state, we get back to the same state as long as we get $T$, so at some point we end up in the state $TH$. Now consider the next two results:
- With probability $\frac14$ we immediately get my pattern.
- With probability $\frac14$ we immediately get your pattern.
- With probability $\frac14$ we get $HH$ and then eventually get your pattern.
- With probability $\frac14$ we get the remaining possibility ($TH$ in the first case, $TT$ in the second case) and either immediately or eventually return to the state $TH$.
Thus, every time we reach the state $TH$, you have twice my chance of winning, so you win with probability $\frac23$ and I win with probability $\frac13$.
By the way, note that this shows that not only the winning probability but also the expected duration of the game depends on the patterns. The only difference between the two cases was that if I choose $HTT$ and you choose $HHT$, we immediately return to $TH$ in the fourth option, whereas if I choose $HTH$ and you choose $HHT$, we eventually return to $TH$ after first reaching $TT$, so the expected duration of the game is longer in the second case.