If I play Frigid Jewel on my turn, and I would draw another card later, does its effect still trigger? Has it already "registered" the card I drew at the start of my turn?

To answer your question, we need to look at how Frigid Jewel's ability identifies the condition and how turns function.

The Short Answer

Yes, Frigid Jewel’s effect will still trigger when you draw your second card later in the turn, provided it is already on the board when that draw happens. It does not "register" or count the card you drew during your mandatory Draw Step at the beginning of your turn; it specifically waits for the second card you draw in a turn.

The Breakdown

1. Timing of the Draw: Every player draws one card during the Draw Step of their Beginning Phase. This is your first card drawn for the turn.
2. Trigger Condition: Frigid Jewel specifically triggers "When you draw your second card each turn."
3. Interaction: Since the card drawn during your Beginning Phase was your first card, Frigid Jewel is still waiting for the next card draw to satisfy its condition.
   • If you play Frigid Jewel on your turn, it enters the battlefield.
   • Any card draw you perform after it is on the board—which would be your second card drawn for the turn—will trigger its ability.

Important Rules

• Triggered Ability (Rule 383): Frigid Jewel has a triggered ability. According to Rule 383.2.c, the condition of a trigger is evaluated after the event occurs. When you draw your second card, the game checks: "Has the condition 'draw your second card' been met?" Since you have now drawn two cards, the condition is met, and the ability triggers.
• Card Draw (Rule 413.2): Drawing the mandatory card at the start of your turn is a game action. Frigid Jewel looks at the total number of cards drawn in the turn (as per the "In a turn" golden rule in your reference). It does not care that the first draw happened before Frigid Jewel was in play; it only cares that the second draw happens while it is in play to trigger its effect.

Summary: You do not need to worry about your opening draw. Frigid Jewel is looking for the second draw of the turn, and as long as that draw occurs while the Frigid Jewel is active on the board, the effect will trigger.