I believe it is the latter - go to a non-Schengen country for 90+ days.
Having said that, what happens is for now, and it is about to change once ETIAS is implemented, some countries are less stringent in enforcing this rule. I am also an American, and I have travelled in and out countries like Spain and France by car, and Italy by plane, on “over-stayed” tourist visa. Travelling by car, there were no border checks of any kind. Nor were there ones travelling by plane.
However, twice I had to resort to my SEF papers to provide the right to enter and stay in Portugal. Once I was requested to show such a proof by TAP in Heathrow, and the other time - by SEF in Humberto Delgado upon passport control. In both cases, showing my SEF certificate of bio completion was sufficient.
And so, the morale is that you might be asked to present supporting documents that you can legitimately enter and stay in Portugal. However, these do not provide you with an exemption from 90/180 rule within other Schengen countries as technically speaking you are an “irregular resident” until you have a card and should remain only on national territory. Your passport stamps at any point of Schengen entry/exit determine your presence within Schengen.
There is an interesting twist exists to the 90/180 rules that has to do bi-lateral treaties signed prior to formation of Schengen area. Thus you can legally stay in France (and technically - only in France) additional 90 days past your 90-day Schengen limit. In Portugal, it is additional 60 days, and I never investigated other countries. Beware that many “young” border control officers are unaware of their existence, and have supporting documents to direct their attention to this matter. Fortunately, I have not had to resort to “educating” them yet. FYI, I posted details on US-PT bi-lateral treaty here: Moving to Portugal right after pre-approval - #48 by PCERoman
After almost year-and-a-half of living in limbo in Portugal, I stopped worrying about these things… 