'Pip' or 'pimp' means five, so your first counting wasn't wrong.

I have never heard of 'tetherer' as the original poster used; I know it as 'tethera'.   Wikipedia lists more than twenty regional counting variants, of which the textilefolksong site uses the Lincolnshire variant in the head of their web page, Yan Tan Tethera Pethera Pimp Sethera Lethera Hovera Covera Dik Yanadik Tanadik Thetheradik Petheradik Bumpit.

See also the Mudcat thread Yan Tan Tethera - more words in the count?

And what has it to do with textile songs? Textiles are produced from wool generously donated by sheep counted by shepherds. Quite far fetched but you need some kind of thread to connect it to a tradition lest the topic would be reduced to industrial songs.