Given that the first flip is a head, the problem is equivalent to asking, what is the probability of getting exactly $1$ head in $2$ flips. There are two possibilities, that the first flip is a head, and the second is tails, and vice versa. This gives the probability the given answer says.
To address your question more directly, you should use conditional probability. Let $A$ be the event that Alex flips exactly $2$ heads, and $B$ be the event that the first flip is heads.
Based on the sample space you wrote out, note $P(A\cap B)=2/8$, as the two possibilities are $\{HHT,HTH\}$. Also, $P(B)=1/2$, as you listed the four possibilities.
Now use the formula
$$P(A|B)=\frac{P(A\cap B)}{P(B)}.$$
I think your main error is that $\{HHH\}$ does not satisfy the condition that the first flip is a head, and that there are exactly two heads, and thus should not be included when calculating the probability $P(A\cap B)$.