I have a vague recollection of reading somewhere that dance was not originally considered a separate activity, being part of a ritual, or music or whatever.
The DIL entry on dán has it also meaning music and dance.
http://www.dil.ie/results-list.asp?mode=BAS&Fuzzy=0&searchtext=dance&findlet=+&f indcol=&sortField=ID&sortDIR=65602&respage=0&resperpage=10&bhcp=1 That may be the reason.
Rinnce is only used in some places in Ireland - the derivation from rink seems a little strained though.
Of course, both Dance and Rink came from French into English...