A useful picture for splinter: something solid cracks, and what comes away is not a neat block but a long thin piece with a sharp point. Every use of the word keeps that shape. The bit in your finger is one. The way old wood breaks is one. And the small, angry group that leaves a big organisation is one too — it came off the main body, it is much smaller than what it left, and it is sharp about it.
From the word’s history
The packet records the noun as Middle English splenter / splinter, from Middle Dutch splinter, and equivalent to splint plus -er; the verb is recorded as coming from the noun. Splint is a thin flat strip of wood, which is why the shape, not just the breaking, sits at the centre of the word.
資料把名詞記成中古英語 splenter／splinter，來自中古荷蘭語 splinter，拆開是 splint 加 -er。動詞則記成是從名詞來的。splint 本身就是薄薄一條木片，這解釋了為什麼這個字的重點不只是「破掉」，而是破成什麼形狀。
Wiktionary (kaikki extract, source_ref kaikki:3f6f69a190036ae4:331172815 and :331207951)
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