Nothing in the universe is completely still, but our Universe behaves much more like your second option than the first one.
You’re absolutely right that things within the galaxy are not expanding along with the Universe at large, and this is because everything within the galaxy is gravitationally attached to the galaxy as a whole, and is not so easily extracted. At the moment, the force which pushes the Universe to accelerate its expansion (the infamously poorly named Dark Energy) is weaker than the attractive force of gravity, which pulls objects together. This is fortunate for us, because it means our galaxy is not being sheared apart by the expansion of the Universe.
The relative strength of gravity in our Universe ensures that anything that’s gravitationally tied to another object is not doing any drifting away from its companion due to the expansion of the Universe. This holds for any set of objects which are ruled by gravity — the stars within a galaxy to the galaxy, or two stars to each other, or two galaxies to each other.
Now, this is not to say that these objects aren’t moving relative to each other — just that this motion is not driven by the Universe’s expansion. It’s driven entirely by gravity. All the stars in our galaxy are following their own orbits around the center of our galaxy, and these orbits are not always perfect circles, so any two stars may find themselves at slightly different distances if you watch long enough.
The stars in Orion’s belt are no exception. They come with the wonderful names of Alnilam, Alnitak and Mintaka, and sit relatively close to us within our galaxy. For some scale, our Sun is about 30,000 light years from the center of our galaxy. These three stars, by contrast, are 1,340 light years, 817 light years, and 916 light years distant, respectively. And these stars are moving, relative to us; Alnilam is moving directly away from us at about 25.9 kilometers every second (it’s also moving sideways, but it’s the traveling away from us part which might be able to make the star fainter). This translates to 58,000 mph, which in astronomical terms is very, very slow. The other two stars are moving even slower — around 18.5 kilometers per second (~41,000 mph).
Considering that a light year is about 5.8 trillion miles (that’s a five, and then 12 zeros), you’re going to have to watch these stars for a really long time for them to make it even a single light year more distant from us. By my calculation, Alnilam, our fastest-moving star, will need about 11,450 years to travel the 5.8 trillion miles in a light year. That star is already sitting at 1,340 light years from us, so an additional light year changes the distance to that star by less than a tenth of a percent — our eyes won’t notice this change, even if we had the 11 thousand years to wait.
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