The raga as Subbarama Dikshitar has written of it, was known as gauDipantu, a janya of mAyAmALavagauLa, featuring M1. gauLipantu just seems to be a variation of the old name that has since then eclipsed. But despite the following ascending and descending sequences.
Aro : S R1 M1 P N3 S
Avaro : S N3 D1 P M1 G3 R1 S
This is a very different rAgA from the one we call gauLipantu today (that one now has M2, has a lot of phrases in common with gauri, and if there was indeed a separate raga gauripantu at one point, it has been totally assimilated out of existence). The presence of M1 instead of M2 completely changes the character of the raga, although not to the extent of what would happen if a pantuvarALi turned into mAyAmALava gauLa. It expresses more pathos in its bhava. And you can linger on the M1 in a way you simply can't on M2.
What else characterizes the old raga apart from the new?
First up, this raga is more scalar and straightforward. Subbarama Dikshitar mentions dhattu phrases like RPMP and vakra phrases like RMDP, SRNDP and PMDMGRS. There is also a RGRS. Other phrases from the kriti of Dikshitar in the SSP and SD's notated raga phrases are DMNDM, DPMPDMMP. Also GRR stands out. R has kampita on it (like it's modern counterpart). SRPMGR, DMGR (both plain and with orikai), DPSN, DPNS, MDPMGR, NN (with the janTa), SNN and even RNDMGRS. In the Dikshitar school, the GRGRSN phrase is absent, although in theory, it could be there via RGRS.
It is also a heavy raga. The M1 and N3 are both capable of long oscillation, featuring gamakas like orikai, nokku and jhArus. SNDPMGRS is indicated with orikai on every note (orikai is a gamaka that goes like S N(S) D(N) P(D) M(P) G(M) R(G) S), touching the upper note as virtual. This even happens in S-P varjya (excluding) phrases.
It is possible, as I suggested in the thread on gauLipantu, that an overly high oscillation of M1 rendered inaccurately might have given rise to it transitioning into M2 - I have the recordings to show it is all too easy. It is also possible that over time, M2 was introduced just for contrast and then it became defacto, but I'm speculating here.
Subbarama Dikshitar calls this rAgA as bhAshAnga, even though it does not have any foreign notes apart from its parent mEla. This labelling has been done for some rAgAs that originally migrated into then CM from other regions and it does not go strictly by the presence of a foreign note.
One characteristic of this raga that seems to have carried over to the present day is that it is middle and lower octave heavy, and does not go beyond upper G in RGRS. So RGRS assumes more importance in the tara sthAyi.
The sole Muthuswami Dikshitar kriti in this raga is krshNananda mukunda murarE (yeah, if that reminds you of someone, it was not MKT who first sang a song with similar lyrics, but MD!!) :
https://www.youtube.com/watch?v=DTZvigJpcd4 - R Vedavalli
https://www.youtube.com/watch?v=3IKwpOFAPco - Mani Krishnaswamy
For a complete SSP baroque version, I am waiting for you, Dr. TRA, when you are done with Thyagaraja (who needs some overdue attention too...)

In the meantime, I will just post the jnAnArnava project's rendition of this kriti here : https://www.youtube.com/watch?v=ASxkJHFcUGw
Phrases like DPMGR, the occasional SNN and SRMPNS and RGRS are common between gaudipantu and gauri, although gauri uses SNN like it's signature phrase, while gauDipantu uses it rarely. The handling of the DPMGR section is similar in both gauDipantu, gauri and sAvEri, so from a musciological point of view, the 3 are part of a family of ragas, with gauLa sharing the SRMPNS with gauri and gauLipantu.
pADi is also rather close, but lacking G, it's phrases are quite different and it is not part of the trio.
There is an older version of Thyagaraja's tera teeyaga rAdA that is still played by the Parur school with shuddha madhyamam. I have shared it in the other thread, but I have felt it deserves it's own spot here. With the additional GRGRSN phrase, this version has almost all the phrases mentioned by Subbarama Dikshitar for gauDipantu, including a NSRGRSN, which is also possible extended from the RGRS phrase.
https://www.youtube.com/watch?v=n3lwl9-3NmY&t=51s - tera teeyaga rAdA seems to actually plead with the Lord of the Tirupati hills with the M1, in a way the M2 version simply doesn't. The M2 version by comparison actually feels angry.

gauDipantu actually should be resurrected on stage again. It is simpler with a lot of scope and it would have been lost in time for good had Subbarama Dikshitar not published the SSP. Originally all the kritis of the trinity were composed in this version of the rAgA. I am not sure about the kshetrayya padams - since we do not know who set them to tune before Dhanammal, one of two possibilities are likely. -- A) They were in another raga called gauripantu (whether this was a separate raga at all is not known) or B) gauLipantu had changed to M2 before these padams were set to their present tunes and gauripantu was just the old name for that version.