The section Robert posted of the AoS solves everything, actually. There is now a Royal House and a Royal Family. Only princes and princesses of the Royal House must adhere to both religion and vicinity requirements. Madeleine and Carl-Philip's children will have the best of both worlds; prince/princess and ducal titles and membership to the Royal Family, but so long as they are Lutheran, they'll not lose their place in succession. Heaven forbid anything happen requiring it, but it looks as though they could fly right in from wherever (still Miami?) in 20 years and take their place on the throne if for some reason it is necessary.
Now, would the people of Sweden like that? Probably not, but if they are at least seen in Sweden and embracing Swedish culture, I bet it all works out fine overall.
As for precedence, etc. Martha-Louise is a great example. She seems to navigate it just fine and doesn't ever seem demoted or lower than anyone else and she still receives plenty of invitations to things.
Has there been any official clarification on the definition of that phrase?
To me the children are being brought up here in South Florida, clearly not Sweden. Or does the phrase refer to education only and therefore Princess Madeleine will return to Sweden so her children can be educated there and remain in the line of succession?
It is open for interpretation. When the Act of Succession was written in 1810, they were afraid to have a King, who had been brought up abroad, maybe at a Royal Court in an enemy country. They couldn't anticipate that it would be possible to fly to other countries in a shorter time than it would take to go from Stockholm to Gotherburg back then.
An interesting thing is that §4 in the AoS is rather muddy. It says:
"In accordance with the express provision of Article 2 of
the Instrument of Government of 1809 that The King shall
always profess the pure evangelical faith, as adopted and
explained in the unaltered Confession of Augsburg and in the
Resolution of the Uppsala Meeting of the year 1593, princes
and princesses of the Royal House shall be brought up in that
same faith and within the Realm. Any member of the Royal
Family not professing this faith shall be excluded from all
rights of succession. "
As you can see the act states that the Dynasts shall be brought up in the Lutheran faith and within the Realm. But the next sentence only states that any member of the Royal Family not prefessing this faith shall be excluded from all rights of succession. In that sentence, nothing is said about where the children are brought up.