From LSJ (here
διάλεκτ-ος , ἡ,
A.discourse, conversation, Hp.Art.30; “θεοῖς πρὸς ἀνθρώπους” Pl.Smp.203a; discussion, debate, argument, Id.Tht.146b; opp. ἔρις, Id.R.454a.
2. common language, talk, “δ. ἡ πρὸς ἀλλήλους” Arist.Po.1449a26; “ἡ εἰωθυῖα δ.” Id.Rh. 1404b24.
II. speech, language, Ar.Fr.685; “καινὴν δ. λαλῶν” Antiph. 171; δ. ἀμνίου, opp. τὰ ἔνδον δράκοντος, Hermipp.3; articulate speech, language, opp. φωνή, Arist.HA535a28; “τοῦ ἀνθρώπου μία φωνή, ἀλλὰ διάλεκτοι πολλαί” Id.Pr.895a6; but also, spoken, opp. written language, D.H.Comp.11.
2. the language of a country, Plb.1.80.6, D.S.5.6, etc.: esp. dialect, as Ionic, Attic, etc., Diog.Bab.Stoic.3.213, D.H.Comp.3, S.E.M.1.59, Hdn.Gr.2.932; also, local word or expression, Plu.Alex.31.
III. way of speaking, accent, D.37.55.
2. pl., modes of expression, Epicur.Ep.1p.24U.
IV. style, πανηγυρική, ποιητικὴ δ., D.H.Comp.23,21: esp. poetical diction, Phld.Po. 2 Fr.33, al.
V. of musical instruments, quality, 'idiom', Arist. de An.420b8.
So either understanding of Papias has support; guess only way to answer it would be to see if there's anything else from Papias that uses διάλεκτος in the same sort of way.
As for your 'best' requests...
I've only found one source that isn't from a fringe-group of loonies: http://markhaughwout.com/Bible/Matthew_Hebrew.htm
I'll let someone who knows the subject better answer the 'best' requests.
My 2c: I prefer Carlson's understanding. I would more expect to see γλῶσσα rather than διάλεκτος, with διάλεκτος to me having more a reference to spoken language, rather than written language.
Though, obviously, γλῶσσα and διάλεκτος overlap very much