Since buttons now support icons from a DO icon set, is there also a way to load DOpus own internal icons?. I mean, the ones you can already use in markup text controls.
I like these icons 
Yes that should work, e.g. #copy should load the copy icon.
If for example, I want to use the 'refresh/sync' icon, #<%oned:4> doesn't seems to works as is.
What would be the correct syntax there?
You're moving the goalposts since those icons aren't from an icon set 
I don't think there's any way to get at those icons at the moment.
I usually refer to these icons as ' DO own internal icons'.
Do they have another name?
They're internal, but they're not an "icon set", as that has a specific meaning.