The Achamana mantra in question seems to have originated from the Padma Purana but its not explicitly mentioned there as Achamana mantras.This is to be noted.
Another noteworthy point is that a common feature in most Achamanas that are found in the Smrithis/Puranas is that they all use Vaishnava mantras.
Quoting from this page:
....As the churning started, innumerable of the proud Danavas who
opted for the head of thousand faced Vasuki out of pride got perished
due to the poisonous flames where as Devas were at the tail end.
Eventally, a massive ‘Halahala’ or the sky-rocketing poisonous flames
engulfed the Univerese and by the grace of Three Names of Lord Vishnu
viz. Om Achyutaya Namah, Om Anantaya Namah and Om Govindaya Namah,
Bhagavan Shankara gulped the blue poisonous flames in his throat thus
giving him the epithet of Neela Kantha. By the power of the three
names of Vishnu, there would never be any hardship faced:
Achyutaananta Govinda iti naamatrayam Maha Mantram hareh/ yo
japetpriyato Bhaktaayaa Pranavaadhyam namontakam/ Tasya Mrityu bhayam
naasti visharogaagnijam mahat/ Naamatrayam Maha Mantramjapedyam
prayatatmavaan/ Kaala Mrityu bhayam chaapi tasya naasti kimanyatah
(Whoever recites the Three Names viz. Om Achyutaaya, Om Anantaaya, and
Om Govindaaya sincerely and with dedication, none of the poisonous
diseases or Agni- based frights/deaths would affect)!
The following image is from an abridged version of Yajanavalkya Smriti.
- He shall perform ācamana thrice and Unmarjana (wiping off) twice. He shall touch the mouth and other parts of the body etc. The water
shall be undisturbed and free from bubbles. This rite is called anga
sparsa and consists of reciting the 12 names of Vishnu while touching
certain parts of the body
So,the Achamana in question seems to be mentioned in the aforesaid Smriti as well.Here Angasparsha refers to the touching of various body parts with the 12 names of Lord Vishnu viz- Keshava,Narayana,Madhava,Govinda ....etc which follow the three sips and & thus completes the process .
In what follows,i'm attempting a compilation of a host(if not all) Achamana Mantras collected from varied sources followed by different Veda Shakhas, Sects etc.
1)3 sips with
the Gayatri Mantra
(from Narada Purana)
2)3 sips with the Mantras
Om Keshavaya Swaha,Narayana Swaha,Govindaya Swaha
(from the Devi Bhagwatam)
3)3 sips with another Vishnu Mantra
"Om apavitrah pavitro va...."
With pre-collected water in a shankha or conchshell or atleast in a
bowl of Kushaa, then perform three ‘Aachamanas’ or sips of spoonful of
water while reciting the Mantra:
Apavitrah pavitrovaa sarvaavasthaangatopiva,
Yassmaret Pundareekaahsham sa bahyaantaraassuchih/
This would be followed by the Aghamarshana Mantras viz....
(from Linga Purana)
4)The following mantras are used by Paipladins(a Shaka of Atharvaveda):
Om Pranaya Swaha,Apanaya Swaha,Vyanaya Swaha,
5)Whereas for the Saunaka Shaka(of Atharva Veda) mantras are as below:
Om Jiva stha jivyasaM sarvamayurjivyasaM
Om Upajiva sthaop jivyasaM sarvamayurjivyasaM
Om Sanjiva stha Sam jivyasaM sarvamayurjivyasaM
6)Sri Vidya Tatva achamana :
Om Atma tatvam shodyami swaha,Vidya tatvam shodayami swaha,Siva tatvam
shodayami swaha& Sarva tatvam shodayami swaha
are the mantras used here.
7)For Chinnamasta (or Nila Sarswati Sandhya Prayoga etc) Achamana Mantras are as follows:
Om atma tatvaya swaha,Vidya tatvaya Swaha,Siva tatvaya swaha.
8)The followers of Arya Samaj use the following mantras:
Sip water from the right palm 3 times after each mantra; Om amrito
pas-taranam-asi swaahaa. Om amrita-pidhaanam-asi swaahaa. Om Satyam
yashah shreer mayi shreeh shra-ya-taam swaahaa.
The above Mantras should be from Vedas and Samhitas(because Arya Samaj gives only Vedas the status of Hindu Shastra & no other).
9)And yet another one(Not sure which sampradaya/sect/shakha use it but the mantras used are from Atharva Veda):
3 Sips with the mantra:
Om Sham no devir abhistaya apo bhavantu pitaye sham yor abhi sravantu
followed by Angasparsha with mantras like :
Vag Vak ,Pranah Pranaha,Chaksush Chaksush,Shrotram Shrotram....etc
The purpose of posting all these(apart from a compilation attempt) is to show that there is no reason to believe that the Achamana given in the question is the Sruti Achamana or that it is the original one(as stated in the title of the question).