M: Looking online to try and get something particular regarding dental floss. It's a fork that has the reservoir, like the entire roll in the handle. But I can never remember what it's actually called to look it up.
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G: Yep. The generic name you're looking for is a refillable dental floss holder or refillable floss pick/handle with built-in spool.
And I found exactly the design you mean: a fork/Y-shaped flosser where the handle itself contains the roll of floss, rather than one where you manually attach a loose piece each time. That design is explicitly described as a floss holder with a spool housed in the handle. citeturn0search3turn0search12
A current example is this one—the wheel advances fresh floss and the handle contains a 66-foot replaceable spool:
product["turn0product1","The Floss Advances - Dental Floss Holder, Tension Adjustable",{"render_as":"hero"}]
For future searches, the phrases that seem to cut through all the disposable floss-pick garbage are “refillable floss holder built-in spool”, “floss holder with spool in handle,” or “reusable floss pick full roll.”
There's also a particularly literal version being sold as a “Reusable Floss Stick – Holds Full Roll of Floss”—it takes an entire ordinary floss spool in the handle rather than proprietary refills. citeturn0search13
So “refillable floss holder” is probably the term you've been trying to remember; adding “built-in spool” gets you the particular subtype.