If I first play a Sprite Mother and a sprite is spawned, and then I follow by playing a Darius, which card is seen as the second card being played, the Sprite token or Darius? Can Darius enter ready in this case?

Yes — Darius is the second card, not the Sprite token.

Tokens aren't cards (Rule 350.2), so playing the Sprite doesn't consume a "card played" slot. Sequence: Sprite Mother is card #1, the Sprite token it makes isn't a card, and Darius is card #2 — so Darius, Trifarian's "when you play your second card" trigger satisfies and he readies (plus +2 might this turn). Same reasoning applies to Gold tokens (FAQ #8290).

Worth noting: Sprite Mother's "When you play me" ability goes on the chain and can be reacted to, but that doesn't change the card count either — the token is never a card.