Ballet's book was not a printed edition, and there're some differences from Playford. The tune (without words at all) from Ballet is quoted in Chappell's National English Airs. Playford may not have quoted the tune from Ballet. Again from John Forrest's book (pp. 307-8):

The dance notation [of "Staines Morris"] first appeared in 1651 (Playford 1651, 87) and was included in the first three editions of The Dancing Master (i.e., 1651, 1655, and 1665), but with the major reorganization of the collection for the fourth edition in 1670, Playford dropped it. The popularity of the tune itself, independent of dance notation, however, spanned a much longer time (see Ward 1986, 299, 303-5). It may first be found in two manuscript lute book of the late Elizabethan era (c 1595-1600), The Trumbull Lute Book (Berkshire Record Office MS D/ED C1 f.9v), and William Ballet's Lute Book (Trinity College, Dublin MS D.1.21/1.f.91), and last shows up in our period of record in two versions in Daniel Wright's An Extraordinary Collection of Pleasant & Merry Humours, Never Before Published (Wright c 1713, Nos. 3 and 13), whence it leaves popular tradition until the morris revivals of the late nineteenth century.