performer gets written to the ID3 IPLS frame (involved people list, for ID3v2.3) or TMCL (musicians credits lists, for ID3v2.4).
In MusicBee you should see the values of these frames in the “Edit” dialog of a track when you click on the three dots button next to the artist name:
foobar2000 I am unsure what the current state is, but AFAIK it does not really support this. It kind of tries to read the data as “<MUSICIAN CREDITS>”, but fails to process the field properly (it is supposed to contain multiple values, where each odd value is an instrument and each even field an artist name). When I last tested this foobar2000 only showed the very first value (which usually is an instrument / vocal name or a generic “performer”). foobar2000 IMHO in general has a bit odd relationship to the ID3 standard 
The internal tag name known to Picard is performer. This is what Picard interprets as representing the performers and knows how to map to different formats and to the MusicBrainz metadata. If you use PERFORMER you create a custom tag. In ID3 this will be stored as a custom text frame as TXXX:PERFORMER.
For ID3 you could also explicitly force Picard to write an arbitrary TXXX field with some scripting. E.g. this would create TXXX:performer with the values of performer.
$setmulti(_id3:TXXX:performer,%performer%)