The sources
The Old English Rune Poem
The manuscript that held this poem, Cotton MS Otho B.x, perished in the fire of 1731. The text we have comes from George Hickes, who printed it in his Linguarum Veterum Septentrionalium Thesaurus in 1705, twenty-six years before the fire. That printed page is now the only witness. Every modern edition, Dickins and Halsall included, is an edition of Hickes rather than of the manuscript.
The manuscript was late and the poem is not. On linguistic grounds Dickins puts the manuscript no earlier than the eleventh century, and the poem itself pre-Alfredian at least, because it does not contain a single definite article. One missing word moves the poem back two centuries. The stanzas on this site are labelled tenth century, which is the conventional dating, and Dickins argues for older.
Hickes printed variant rune forms alongside the poem, and Hempl showed they came from a different manuscript, Domitian A.ix. That is where the wrong names come from: in the 1705 edition Day is labelled Man and Man is labelled Day. The poem never names its own runes, so every name above every stanza in every edition was put there by a later hand.
Twenty-nine stanzas of two to five lines. Eight runes have no Norse stanza at all, so for those eight this poem is the only old source that speaks.