Short answer
If a chart or jeweller pack says 376 or 377, you are in the SR626SW silver-oxide family (~6.8 × 2.6 mm) — not choosing between two rival sizes.
Do not treat “376 vs 377” as the alkaline vs silver decision. That decision is LR626 / AG4 vs SR626SW / 377 — see LR626 vs SR626SW.
What the numbers mean
377 / 377A / D377 / V377 are common retail prints for SR626SW. 376 / D376 appear on the same size class in many trade tables (including this site’s watch battery chart).
Buy silver-oxide SR626SW when the movement or empty cell names 376 / 377 / SR626SW. Confirm the chemistry printed on the blister before checkout.
The real mix-up
Alkaline LR626 / AG4 is the same physical size but a different chemistry. Charts that print “AG4 = 377 = 376” as one battery collapse that distinction — we keep them separate.
Do not use list
Do not buy alkaline AG4 / LR626 when the watch asks for 377 / SR626SW just because a chart listed AG4 next to 376 / 377.
Do not confuse this pair with thinner 364 / SR621SW (~2.1 mm) cells.
Next steps on this site
Open LR626 vs SR626SW for the alkaline vs silver-oxide answer, and the watch battery chart for other trade numbers.