tolkiens legendarium - What was the first sentient race that awoke on Arda?

Some background: Tolkien's history of elves singing and ents walking and talking.

One of the foundations of Tolkien's mythology is the Elves being the Firstborn. The Dwarves are even put back to sleep until the Elves awake. I've always taken this to mean the first born sentient race¹.

However, several quotes are making me question this:

Even some whole inventions like the remarkable Ents, oldest of living rational creatures, Shepherds of the Trees, are omitted.

^{The Letters of J.R.R. Tolkien, To Milton Waldman (#131), p. 160}

Ents. The most ancient people surviving in the Third Age were the Onodrim or Enyd. Ent was the form of their name in the language of Rohan.

^{The Lord of the Rings: Return of the King, Appendix F: The Languages and Peoples of the Third Age, p. 511}

[...]before the Children awake there shall go forth with wings like the wind the Eagles of the Lords of the West."

^{The Silmarillion, Of Aulë and Yavanna, p. 41}

There's another quote from the Silmarillion abouts Ents that contradicts the ones above:

When the Children awake, then the thought of Yavanna will awake also, and it will summon spirits from afar[...]

^ibid.

Concerning the Dwarves, Ilúvatar says this:

But I will not suffer this: that these should come before the Firstborn of my design, nor that thy impatience should be rewarded. They shall sleep now in the darkness under stone, and shall not come forth until the Firstborn have awakened upon Earth

^{The Silmarillion, Of Aulë and Yavanna, p. 38}

So, are Elves not the first sentient beings to awaken? Were they only called "Firstborn" in relation to Men? Was it the Ents? Or maybe Eagles?

I am obviously excluding from this any race that was created outside Arda (Ainur, Maiar) or weird individuals (Bombadil, Ungoliant).

^{1"race", "species", "ethnicity" or whatever. Not trying to start a semantics war.}

^{All citations from the Harper Collins editions}