The question involves understanding the bi-implication completely.
The key statement is
The result of the toss is head if and only if I am telling the truth.
Now let us denote two propositional variables p and q where
p: "The result of the toss is head"
q: "I am telling the truth"
and hence the statement is $p\Leftrightarrow q$
Now we have two cases
Case 1: The person is of Type 1: Here it is very simple case.Since Type 1 person always tells the truth and hence according to him the result of the toss is Head.
Case 2: If the person if Type 2, then he always tells lies. Means, the statement $p \Leftrightarrow q$ given by this person is actually the negation of the original statement which he means
i.e. $\sim (p \leftrightarrow q) = (p \leftrightarrow \sim q)$
so $p \leftrightarrow \sim q$ is The result of the toss is head if and only if I am not telling the truth.
And, yes this bi-implication is true for this Type 2 person because he never tells truth. So, here in this case also the result is Head.
And hence, the result Head is consistent irrespective of the type of person we choose.