You most definitely have the correct ideas/understandings.
But I find the notation very, very awkward. Remember that EVENTS are subsets of the probability space. The elements of each event are ELEMENTARY EVENTS which are in this case are pairs, the outcome on the coin and the outcome on the die.
So I would do this:
$ \begin{align*}E_1&=\{(t,~)\} \\E_2&=\{(~,4)\} \\E_3&=\{(h,2),(h,4),(h,6)\} \end{align*}$