Sanskrit term for a "Buddhist"

I'm not a Sanskrit scholar but a quick dictionary search gives the following:

bhikṣu - a beggar/wanderer
śramaṇa - a spiritual seeker
pravrājita - the one gone forth
anusārī - follower

The problem is, back in Buddha's times there were hundreds of different sects, and every seeker either went from one sect to another and assembled their own dharma from the elements available on the spiritual market - or picked one teacher to follow. In the first case they were simply called a bhikshu or shramana, and in the second case they were called a follower of such and such teacher (e.g. of Shramana Gautama, "sramanagautamanusari").

So the closest to "Buddhist" would probably be "buddha-pravrājita" or "buddha-anusārī"

The term Dharmānusārin would mean "one who follows in the Dharma" in Sanskrit. Please see the definition in this dictionary entry.

The term Śraddhānusārin would mean "one who follows in faith" in Sanskrit. Please see the definition in this dictionary entry.

The words Dharma and Śraddhā can be merged with anusārin according to sandhi rules.

On this page, from S.N. Goenka's Keynote address at the International Bauddha Mahotsav (Buddhist Conference), at Sarnath, India in 1998:

If I had been aware, much earlier, that this conference was named as 'Bauddha Mahotsav', then I would have suggested that a more appropriate name would be 'Buddha Mahotsav', or 'Dhamma Mahotsav', or even 'Tiratana Mahotsav'. This is because in the entire ancient literature of the Buddha's teachings, commentaries and sub-commentaries, amongst a total of 59,150 pages containing 9,285,755 words, the word 'Bauddha' is very conspicuously absent. The Buddha never taught 'Buddhism' or 'Bauddh Dharma'. He never taught a religion, nor did he convert anyone to an organized religion. He taught Dhamma (Dharma) - the universal laws of nature - and inspired people to follow Dhamma.

Those who followed the teachings of the Buddha were never called 'Bauddh' (Buddhist) during the time of the Buddha, and even till about 500 years later. Throughout this huge literature, we find only the following words referring to those practicing the Buddha's teachings: Dhammi (Dharmi), Dhammiko (Dharmika), Dhammattho (Dharmastha), Dhammacari (Dharmacari), Dhammavihari (Dharmavihari), Dhammanusari (Dharmanusari).

We don't know when, where, by whom and why the use of the word 'Bauddh' first began, and so also the use of the words 'Buddhism' and 'Buddhist'. When we say that the teachings of the Buddha are 'Bauddh Dharama' (or Buddhism), and say his followers are Bauddh (or Buddhists), obviously it means that the teachings are meant only for people claiming to be 'Bauddha' or 'Buddhists'. Whereas the Buddha declared so emphatically that Dhamma is infinite - "appamano dhammo".

When the teachings are called Buddhist, or ‘Bauddh Dharma' or Buddhism, then most of the suffering people of the world will get frightened thinking that they are being converted from one particular religion to another, and be deprived of the universal teaching. But when Buddha's teachings are given under the original, true nomenclature of ‘Dhamma', and not any other terminology, then people are reassured of the basic fact that the Buddha's teachings are not just meant for any one religion. So people from any background will have no hesitation accepting his teachings.

Our research at Igatpuri has shown us that till about 500 years after Buddha, the word ‘Bauddh' was not found in any ancient spiritual literature of India-the literature in the Buddha tradition, Mahavir tradition, or Vedic tradition. The distinguished scholars assembled here may help us make a proper research as to when usage of this word ‘Bauddh' began.

It is most important to remember that the Buddha's teachings are not meant to be limited to any particular sect. His teachings are universal. And if we see the Enlightened One's teachings in proper perspective, it is clear that the original teachings of the Buddha are totally universal and non-sectarian.

In pāli canon, follower is sāvaka (su+ika=follower to hear dhamma from satthā [saha+attha=philosopher who has philosophy in his mind]).

Sāvaka, in pāli, is śrāvaka, in sanskrit.

This word use in every canon. You can put "buddha" word in font of "sāvaka", buddha-sāvaka, to specify just a buddhist person.

https://www.buddhistdoor.org/tc/dictionary/details/savaka

P.s. use this tool to convert between thai and roman of the first reference link.