The horse is the jumper because it jumps from one space to the next. And because horses can jump.
Additionally for the other two pieces with significantly different names in the article:
the rook is called tower so the shape obviously remains the same
the bishop is called runner/walker because unlike the jumper, he cannot jump. And he's not a tower, so clearly there's no other reasonable name for him.
It's so easy to tell this map was made by a Brit. Wales gets its own color (despite largely not speaking Welsh) but Belgium and Switzerland are monochrome (despite having multiple federally recognized and geographically partitioned monolinguistic regions and their own flavors of historical-but-rarely-spoken language)?
Only the Bri'ish would be haughty enough to assume their flavour of federal governance is so unique.
(I don't actually care, it's just very interesting how even such an innocent map actually shows a strong political/cultural bias)