In comments, BillJ wrote:
The adverb "only" is a focusing modifier. In 1. it modifies the VP "dream of becoming someone like Dr. Stone", and in 2. it modifies the VP "do this for so long".
I know what you mean. "Only" isn't negative, but it is semantically close to a negative in that "Only John liked it", for example, entails "No one other than John liked it". I think your interpretations are acceptable too.
In comments, John Lawler wrote:
It doesn't represent anything. It's not referential; it's a quantifier, with a focus, as @Billj said.
That's the ambiguity of the focus. Only can appear modifying any constituent that contains its focussed element; here they are verb phrases with several possible foci. That would be clear in speech from the stress; but in writing you can't tell what word's stressed, so writers are advised to put only right before its focus if they want to avoid ambiguity.
Only is a negative trigger, though a weak one. It also involves presupposition and quantification, so it's a regular mare's nest. Larry Horn wrote the classic analysis paper around 50 years ago.