The most simple method I was always taught was when giving a definition or further explanation to a term to use ~という意味だ, ということだ, or という意味がある
For example,
「名所は特に有名な所ということです」
「武士道は侍の尊敬と礼儀を表し方ということです」
I know that there are other methods out there, but it seems that for what you're trying to do this might be the simplest and most direct path.
As the other response mentioned, 含む (to include) might be a useful verb to have on hand here to suggest that when discussing sake that meaning is already included.
To directly translate the example that you were discussing, I might say something like this: 「酒はアルコールが入っている飲み物という意味がありますが、ある時に酒の意味に日本酒ということも含みます。」
AはBを意味する
for sort of direct equivalences, whileAはBの意味を持つ
would also have a sort of implication that A also means some other stuff, but I don't fully understand the nuances between these and other options, so I'll let someone else answer.