[Ahhh, this one. Okay, there is a very long story associated with this, including an entire essay written by Tolkien about the pronunciation of this sound. People have written better, and fuller, explanations of it than I have, and I admit to being no real Quenya expert at all.
But I’ll try to explain it briefly. The various elvish languages contain a sound often written with the character called a “thorn,” which would look like this: Þ (The symbol thing I use, yes?) This sound is known as the voiceless dental non-sibilant fricative, as opposed to the voiced dental non-sibilant fricative. (In English, compare the words thorn and this. The initial “th” sounds are subtly different. Say them slowly, holding your windpipe as you do so, and you’ll be able to feel the difference.)
In Noldorin Quenya, the voiceless dental non-sibilant fricative began to undergo a mutation after the birth of Miriel Therindë and before the birth of Fëanaro, becoming instead an S sound. However, Miriel preferred the original pronunciation of her name, Therindë rather than Serindë, and all her kin used it because of her wishes — even after her death, Fëanaro insisted that the Þ sound continue to be used, viewing rejection of the Þ as a rejection of his mother herself.
This is known as the Shibboleth of Fëanor — a shibboleth being a preferred pronunciation, basically. All of his sons kept to this shibboleth for his sake and that of his mother. So Caranthir will always use it in words which should take it.
Okay, so there’s the background on Th vs. S. Now we come to the Sindar or Thindar question.
When the exilic Noldor first arrived in Beleriand, the elves later known as the Sindar did not have a name for their entire people, other than eledhrim, which just means elves.
They instead used names to describe individual populations of their folk, like the Iathrim…the people living in Doriath. If I understand it correctly, until a brand-new and very different population of elves arrived in their lands, there had never really been a reason to establish a name differentiating their own “ethnic group” from another, if that makes sense?
But then the Noldor arrived and they named the elves living in Beleriand, the Sindar, the grey-elves, to distinguish them from their own folk and from the green-elves, etc.
So Sindar and Sindarin are, in fact, words in Quenya, not in the tongue of the elves they describe, the language commonly called Sindarin! These words are therefore, in fact, likely not what the grey-elves called themselves amongst themselves.
So since the Fëanorians use the Þ sound, they would say Thinda and Thindarin, while other populations of Quenya speakers would say Sinda and Sindarin. But the in-universe conceit of the Silmarillion is that it is a history later written down….by people who did not hold to the Shibboleth of Fëanor, and therefore used the mutated S sound.
Fun fact, though: Even if we decide to assume that the Sindar/Thindar adopted that name for themselves due to convenience (let’s ignore Thingol’s Quenya ban in so doing), Sindarin actually preserves the thorn!
See Thingol, as opposed to the non-Shibboleth Quenya name Singollo, for example. The element thin- in both Thingol and Thindarin comes from the same root, meaning grey.
So with a couple assumptions made, there’s a pretty good chance that after the Noldor arrived, the Sindar might have called themselves, yes, the Thindar.