The symmetry of this problem provides another way to look at it:
you choose a coin at random and look at one of its faces at random.
That face shows something (maybe heads, maybe tails).
What is the probability that the other face of the coin is the same as the
face you can see?
The answer to that question is clearly $\frac23$, since two of the three
coins have the same thing on both faces and the third coin has something
different on the other face no matter which face you look at.
In which case is the probability greater that the other
face is the same as the face you can see: (A) you see heads;
(B) you see tails; or are the probabilities the same?
By symmetry (because the conditions of the experiment are exactly
the same even if we relabel every "heads" as "tails" and every "tails"
as "heads"), neither probability is greater; they are the same.
So it doesn't matter which face you see when you first look at the coin;
there is a $\frac23$ probability the other face is the same,
even if the first face is heads (as it is in the original question).