I've found a few indications after a brief perusal of the Digest. Please not that this is currently a collation, rather than an actual study - there are works on testaments out there, which it might be useful to check, and which will have gone into much greater detail. The Digest tends to go off into quite specific cases, which can be daunting to non-lawyers.
At any rate, the bits concerning your question seem to be in Books 28 and 29 of the Digest. 28 largely concerns the question who can make a testament (e.g. you can’t if you’re deaf or mute, insane, in the power of the enemy...), and for what reasons it can be declared void (always useful to know if you have been disinherited); it also includes the methods of disinheriting, largely under 28.2. A major part here is concerned with posthumous heirs, but also with formulae regarding the disinheriting.
Working from a French translation of the Digest at
http://www.histoiredudroit.fr/corpus_iuris_civilis.html :
It seems usual to disinherit “by name” (
nominatim exheredari), but that is not actually necessary so long as it is clear who is meant. 28.2.2 = Ulpian Regulae VI says that one can disinherit a son by the words: “
filius meus exheres esto”, or “let my son be disinherited”, without actually naming him. This works only if there is only one son. He does not say whether this is a purely verbal action (presumably in front of witnesses), or whether it needs to be recorded. Florentinus, Institutiones X = Dig.28.2.17 adds the formulae “
filius exheres sit” and “
filius exheres erit” (let my son be/my son will be disinherited).
28.2.26 = Paul, Sententiae II mentions a decree of Augustus which forbade fathers from disinheriting a son in the army, but he also mentions that this decree had been abrogated; he does not mention by whom it was abrogated.
28.2.14 = Africanus Quaestones IV, for instance, says that if a father disinherits a son by writing (the word here,
scripserit, is clear about a written form) “let him, who I know not to be born from me, be disinherited”, and the son can prove that the father was mistaken – however that is going to work without knowledge about genetics – the disinheriting is void.
As regards the modalities following which a testament has to be made, it’s quite important that there were witnesses. These witnesses need not even understand Latin (Dig. 28.1.20 = Ulpian ad Sabin. I, quoting a rescript of Marcus Aurelius to Didius Julianus), so long as he can see what is going on: that might be useful, since you ask for the process in the province of Achaea. The texts collected in the Digest largely suggest written testaments, but there are indications that this is not always necessary: Dig. 29.1.24 = Florentinus, Institutes X quotes a rescript from Trajan to Slatilius Severus “... we know that a testament can be made without having been written down, even by those who are not soldiers”, although the text is largely concerned with men in the military.
All that is assuming we are dealing with Roman citizens. I'm not sure what law plays between peregrini or provincials. I would expect the Digest assumes that everybody who is free is also a citizen, considering it postdates the Antoninian Constitution (even if many of its sources don't).