The meaning is the same, both in Brazil and in Portugal.
As a curiosity the two countries (along with others) are part of the Portuguese Language Orthographic Agreement of 1990 [1] whose purpose was to create a unified orthography for the Portuguese language, to be used by all the countries that have Portuguese as their official language.
Nonetheless there is a lot of differences in usage, mostly regarding the "popular" vocabulary. Some heavily used words in one country are less frequently used in the other. There's also a noticeable difference in the accent.
[1] https://en.wikipedia.org/wiki/Portuguese_Language_Orthograph...
Context: I'm Brazilian and during high school I studied Portuguese "one official version behind". It had more rules, but the language itself was way nicer, and people used to write more close to it, than nowadays.
It's better to have a few mistakes due to some archaic rules than to force everything to be sterile and complicate how you pronounce things IMHO
Just let people evolve the language naturally instead...
I can tell you though, that using the word "periferia" is actually common in some place (although a bit rare in most). It's mostly used as a synonym for 'around the borders'.
The example you've mentioned: "move the burgers to the periphery of the barbecue", actually seems perfectly fine. It seems a little bit over-detailed, I guess that a bit context-dependent but I wouldn't bat an eye to the equivalent of "move them to the periphery".
From what I recall 'periferia' is also commonly used in TV news.
The chap in question was from Portugal. If I understand correctly from your website, you're from Brazil, right? Is there any difference between how that word would be used in Brazilian vs European Portuguese?