Difference between Dominant strategy and Winning Strategy
Your intuition seems mostly correct. However don't forget that not all games are winner vs. loser, see e.g. Coordination games. The meaning of a winning strategy is somewhat meaningless there, unless you define the winner to be the one with the greater outcome (or allow multiple winners which gain more than a certain threshold). Dominance is still an important tool for those strategies.

There is another problem, because your definition of domination includes "better" in any case. This is a strict dominance. A weak dominance means "at least as good as" in all cases and "better" in at least one case, where case means an opponents strategy (or a set of strategies for multiple opponents). 


Possible outcomes are only $\{\text{win},\text{lose}\}$: A winning strategy for you results in $\text{win}$ for any strategy of the opponent and is thus weakly dominant compared to any non-winning strategy, if there is one. But in many games an opponent can play so badly, that some non-winning strategies lead to a win nevertheless and so they are not strictly dominated by a winning strategy. But more important than the dominance of winning strategies is the dominance between non-winning strategies of your opponent. If s/he has a weakly dominant strategy relative to all other strategies and you can win against this single strategy, you have a winning strategy.
Possible outcomes are some set linear ordered set $A$ and you win, if your profit is greater than that of the opponent (this is not a usual definition, one would use other tools like equilibria in this case). For example in the following table each player has two strategies and yours correspond to the rows (each cell is: your outcome, others outcome):
$$\matrix{10,12&amp;\mid &amp;10,12\\\hline\\7,4 &amp;\mid&amp;7,4}$$
If you choose the second strategy you'll always win and if you choose the first one, you'll lose, but your outcome is higher, so choosing row $1$ dominates a winning strategy.


Resumé: Dominance is a much broader concept and for many  but not all games winning strategies are weakly dominant to at least one other strategy.