The meanings for the headword and for its first gloss are not attested in Greek texts. Although the prefix meta- does rarely mean 'afterwards, too late' (LSJ s.v. meta/, G.VI and III, web address 1, at the end), it is not known in this sense with these two verbs; they are not, to our knowledge, synonyms of a)f-ustere/w (web address 2).
This error enters Platonic scholarship with Olympiodorus and is perpetuated in Timaeus' Platonic Lexicon (149) and in the lengthy note on the entry by Kühnken in his edition. It hinges on a misunderstanding of the ironic and witty exchange between Callicles and Socrates with which Plato opens his Gorgias (447A: web address 3). Socrates and Chaerephon have arrived late and missed a dialectic exchange. Callicles quips, "They say it is right, Socrates, to participate [the normal meaning of metalagxa/nein] this way in a war or a battle." Socrates catches the drift of the quip and replies, "Aha, the proverb! We have come after a feast and we are too late." The proverb, as reconstructed, must be to the effect, "Be late to a battle (i.e. when it is over) but early to a feast." Shakespeare appears to know the proverb at Henry IV; Part I 4.2, "Well,/ to the latter end of a fray and the beginning of a feast/ fits a dull fighter and a keen guest." The exchange between Pompey and Cicero in the Civil War is probably also based on the proverb. Pompey complains that the other has chosen a side late in the war; Cicero responds to Pompey's remark with the celebrated "I have not come late at all; for I see nothing ready here" (Fac. Dict. 19: Minime sero veni; nam nihil hic paratum video), perhaps a sly allusion to a proverb about those who prefer dinner to war.
The ancient commentators merely misunderstand the ironic exchange when they take ou(/tw metalagxa/nein in Callicles' sentence to be a synonym of Socrates' a)fusterei= in his response. Olympiodorus in his Commentary ad loc. (1.2.5) misquotes Callicles' quip and puts u(sterh=sai in the place of metalagxa/nein. He explains that Callicles meant Socrates had arrived in a war and the company understood it was better for him to have arrived late. Socrates, in his view, then introduces a quite different proverb, to imply that dialectic is not a war but a feast and he should have arrived early. This is not implausible, but it runs afoul of the proper meaning of the headword, which appears the attested reading rather than that of Olympiodorus.
Hesychius mu1003 has the correct meaning of the headword, "to participate" (mete/xein). Photius mu317 Theodoridis gives alternative meanings for the present infinitive of the headword: "to participate or to arrive late or to miss out on a share". The Suda's meth/|ei, an implausible imperfect in an unknown sense of me/teimi ('I go after'), is probably a simple copying error for a form of mete/xein(cf. Anecdota graeca, ed. L. Bachmann 299.9) and should probably be emended (see mu 780 note).
See also mu 786.