Sellenger's Round wasn't in the 1651 edition of The English Dancing Master, it appeared in editions 3 onwards (3-18), with alternate title The Beginning of the World, so 1657 is its first appearance in Playford.

I don't think that Spring Garden is an alterative title. I believe this stems from a misreading of the index in Jeremy Barlow's The Complete Country Dance Tunes from Playford's Dancing Master (1651-1728). The index entry reads:

Sellenger's Round 129 Sellengers Round 132

Then entry is the title and tune number in Barlow ('Sellenger's Round 129') followed by the name it was listed under in the first edition it appeared in and page number (or tune number in editions that didn't have 1 dance per page). So the 2nd part of the index entry 'Sellengers Round 132' means tune was listed as 'Sellengers Round' on page 132 of the 3rd edition. However if the 2nd part of the index is read as another entry in Barlow then tune 132 in Barlow is Spring Garden and I think this is where the mistake comes.

I don't know if the other two versions are more than variants of each other. For example the Playford version is more elaborate than the Byrd's simple statement in Fitzwilliam.

Simpson's BBBM has a long entry on the tune with various sources listed (eg William Ballet's Lute Book, which I think is late 15th/early 16thC). He also says 'References to the tune under one name or another abound in literature of the 16th and 17th centuries'.

Also (from BBBM) the origin is unknown though has been supposed that name derives from 'St.Leger's round'. Grattan Flood argued for an Irish origin and name derived from Sir Anthony St. Leger, lord deputy of Ireland in 16th C. (Though if memory serves Grattan Flood is known to be fanciful of origins! MCP)

Mick