This page is plain wrong. There is no "Old Entish" predating contact with the Elves.
In Appendix F, it is said:
"They [the Ents] were known to the Eldar in ancient days, and to the Eldar indeed the Ents ascribed not their own language but the desire for speech."
Besides, there is no "New Entish" that they would use among themselves.
In Appendix F, it is further said:
"The language that they had made was unlike all others: slow, sonorous, agglomerated, repetitive, indeed long-winded; formed of a multiplicity of vowel-shades and distinctions of tone and quantity which even the lore-masters of the Eldar had not attempted to represent in writing. They used it only among themselves"
This article is mostly a mess of unsupported assertions and inventions.
The Elves taught the Ents the concept of communicating with sounds. Afterwards, they came up with their own language which Tolkien didn't attempt to transliterate. He explains quite clearly in Appendix F that there is a difference between when Ents are using Elvish in Ent fashion, and when they are speaking in pure Old Entish - saying that even the Eldar could not hope to represent it accurately in writing.