SImilar words are often used with quite different reference. It might be helpful when discussing pronunciation to specify the 'kind' of whatever name one puts on a system.
For example, some have referred to reconstructed Koine as 'restored' while others have referred to reconstructed 5c BCE Athenian as 'restored'.
Some refer to 5c BCE as 'Erasmian', while many refer to various local 'mixes' as Erasmian.
In the US, most local Erasmian kinds use fi, theta, chi (fricative, soft consonants), while reconstructed 5c BCE Athenian uses hard, aspirated stops for those sounds. Also in the US, many "Erasmians" pronounce 'o'-mikron like alpha, while others use an "aw" as in 'saw', while those emulating 5c BCE use w-mega for 'aw' and 'o'mikron for a close 'o'.
There is one particular kind of Erasmian that I would like to focus on in this post before suggesting distinguishing names.
Many Erasmian Anglophones follow the advice of Gresham Machen. They pronounce EI the same as H, that is, they pronounce ει the same as η. Now we can be fairly certain that such was not the case in 5c BCE Athens. At the time of Alexander and the spread of the Koine it was only EI that merged with I (iota), not H. (For documentation, see any good historical phonological study. Horrocks provides a good overview. We need to be speaking about post 403 BCE, of course, since the H symbol was originally a consonant and not the long, low, front vowel.) Such a specific sound merger dealing with only half of the pair ει/η could only have happened if EI and H were distinct at the beginning of that process. So we can ask the question, if EI is allowed to be merged with another phoneme in an Erasmian system, to which vowel should it be merged, to H (eta) or to I (iota)? The answer is straightforward. Historically, it was not a part of H, so it would be a poor choice and would misrepresent the Greek system to merge the sounds for EI and H as some kind of mid front vowel [IPA e, ε]. On the other hand EI is known to have merged with I in the third-fourth century BCE when before consonants, and before vowels as well within a couple of centuries. What this means is that the "Machen Erasmian" is just wrong on this vowel and the Machen sound equivalence was unnecessary for English speakers. "Machen Erasmianists" would do everyone a favor if they simply dropped EI= [e] and joined the flow of the Greek language and used EI= [i]. [PS: IPA "e" is the starting sound in English "fate, bait, weight", while IPA "i" is the sound of English "feet, ski, belief, leaf".] If they were to join EI to I they would be able to distinguish indicative clearly from subjunctive in words like βλέπει "s/he sees" versus βλέπῃ "s/he would be seeing". And their resulting dialect would sound more "Greek" to Greeks and to anyone using a phonological system that is updated from "restored 5c BCE" [PS: "Restored 5c BCE" is NOT a standard Erasmian and should not be assumed to be whatever "Erasmian" someone may be familiar with].
While correcting this one item within "Machen Erasmian", one other item should also be introduced that would not affect its overall "Erasmian" nature. Machen Erasmian uses soft, fricative "fi, theta, chi" for the series or originally voiceless aspirated stops. [PS: an aspirated stop is like the "t" in English "tea, tree" as opposed to the fricative in "three, thistle".] Now those voiceless fricatives that have been adopted in Erasmian are from the Koine period of the language. More importantly, those voiceless fricatives apparently developed AFTER the voiced stops β, δ, γ became fricatives. It appears that β and γ became fricatives by the first century CE at the latest. So Machen Erasmians could do everyone a favor be including β, δ, γ, φ, θ, χ as fricatives in their system. That would go a long way to ameliorating the cacophony that is heard in modern Greek ears when a Machen Erasmian attempts to speak. [PS: Greeks endure that at conferences and join the Erasmians themselves in paying attention to the printed text. Erasmians rarely pay attention to clause-length and longer Greek when read orally, they depend on printed handouts/texts, just like in most Greek reading classes.] Anyway, Machen Erasmians could go a long way toward rapprochement with the Greek language by correcting their EI into I [iota sound] and by using fricatives for both [φ θ χ] and [β δ γ].
So finally some names to help with a score card when talking about "Erasmian".
"Restored Attic" should be reserved for 5c BCE Greek where long vowels are produced longer in time, accents like circumflex and acute are distinguished, the consonants [φ θ χ] and [β δ γ] are all stops, and vowels like η and ω are slightly lower than their short counterparts ε and ο. [PS: that is, H is closer to "ae" in "bae, baa" and ω is closer to 'aw' in "saw".
"Machen Erasmian" would be used for a collation of EI with H, fricatives for only φ θ χ, ο/ω reversed as 'o'="aw" in "saw" and ω = 'o' in "note", and normally with no distinction on the kind of accent. This is a very common form of USA Erasmian.
"Corrected Machen" would be Machen Erasmian with the corrections added above: EI would go with I and the stops [φ θ χ] and [β δ γ] would all be fricatives.
"Alpha Erasmian" could refer to a Machen-esque Erasmian where o-mikron is merged with alpha. Some have referred to this as Astartean Erasmian because some phrases like τον θεον "the god" sound like ταν θεαν, which is Doric for "the goddess" (cf. John 1.1).
This, too, is fairly commonly encountered in the USA.
"British Erasmian" might be used as a subtype of Machen Erasmian where the accents are often ignored in reading. [There are various vowel qualities in some of the British Erasmian systems which would further complicate matters.]
"Restored Koine". By way of contrast with "Restored Attic" above which has tried to consistently apply the "Erasmian" principle of a historical reconstruction of one symbol=one sound, "Restored Koine" could refer to reconstructed Koine where the stops [φ θ χ] and [β δ γ] are fricatives, vowel sounds are seven [ι η ε α ο ου υ] or six in later Koine [from the time of Origen: ι ε α ο ου υ], and accents are read as the correct stressed syllable (or sometimes without stress in the case of grave when read fast). There is a new webpage link for a description of all of this: http://www.biblicallanguagecenter.com/w ... n_2008.pdf
We might as well add a note on 'historical pronunciations'. Often, "Historical Pronunciation" is used for Modern Greek with its five vowel system [ι ε α ο ου], fricative consonants, and accents that are read as the correct stressed syllable. Modern Greek was the pronunciation at the time when Greek studies became more widespread in Europe following the fall of Constantiople. 'Modern Greek' is probably the best term for this today since most linguists would consider both Restored Attic and Restored Koine to be historical pronunciations.
The use of some adjective on "Erasmian" will help both Erasmians and non-Erasmians understand what is being talked about.