Edited: Welp, it seems I am wrong!
The Silmarillion, "Index of Names.": It is said about the Eldar:
"According to Elvish legend the name Eldar 'People of the Stars' was given to all the Elves by the Vala Oromë. It came however to be used to refer only to the Elves of the Three Kindreds (Vanyar, Noldor, and Teleri) who set out on the great westward march from Cuiviénen (whether or not they remained in Middle-earth), and to exclude the Avari."
So both the Sindarin and Silvan Elves are indeed considered part of the Eldar.
And here I was thinking I was really smart...nope, just overconfident. But it's all good. I needed a slice of humble pie.
I was researching Tolkien's elves and something popped out at me that I hadn't noticed before.
In the Hobbit, when Tauriel and Kili are swapping stories about stars, I never caught this lore inaccuracy until now. (Ignoring the fact that Tauriel said it. We aren't here to talk about her [non] existence specifically.)
Tauriel says in her little star speech that "all light is sacred to the Eldar, but Wood-Elves love best the light of the stars".
It is true that at their birth in Arda, the Elves found by the Maia Oromë were named "Eldar" as a whole, but that name became reserved for only the elves who completed the journey westward across the sea to Aman, the Blessed Realm. Eldar became synonymous with High-Elves, those who lived under the light of the Two Trees. This name now only applies to the Vanyar, Noldor, and Falmari (Teleri who crossed the sea).
So why is Tauriel, a Silvan Elf (a Wood-Elf branch of the Nandor), calling herself - and her kin by extension - of the Eldar? The Nandor Elves stopped their journey on the eastern side of the Misty Mountain, refusing to cross over because they were afraid of the looming shadow the mountain range cast over the land. Mind you, the only light they had in Middle-earth at the time was starlight because the Sun had not risen in the sky yet.
And to that point, technically not even Elvenking Thranduil can call himself an Elda because he was of the Sindar, born under the rule of Elu Thingol in Doriath. He and his father Oropher never travelled West across the Great Sea. Thranduil never saw the light of the Two Trees, which is the requirement for being an Eldar.
Lore inaccuracy! Gotta love it!
But to her credit, Tauriel was right that the Silvan Elves (Wood-Elves) much prefer the light of the stars because when the Sun rose for the first time, it was like a terrifying ball of fire in the sky with absolutely no warning. The Dark Elves of Middle-earth were forced to become Light Elves after centuries of only starlight. Talk about a rude awakening! ☀️