It made the call “23 skidoo, kid” famous, right?
99% of the time, I would be, and if I wanted to be specific, I’d say “steam iron” or “clothes iron” before I said “flat iron”. Just look at the Wiki article, it uses plain “iron” many, many times.
And a clothes iron’s just as “flat” as a golfing iron, in that the working surface is a plane, grooves & steamholes notwithstanding.
East Wing, National Gallery
http://www.greatbuildings.com/buildings/east_wing_national_gallery.html
Ditto.
New York has “the cast-iron district,” from the construction of the buildings once housing heavy manufacturing machinery; its ambiguous borders are generally considered Soho-ish, but beautiful examples extend uptown to the former grand department stores in the teens.
Flatiron was differentiated from other types of irons in the 19th cenutry: Sad iron or sadiron; Box iron, ironing box, charcoal iron, ox-tongue iron or slug iron; Goose, tailor’s goose or, in Scottish, gusing iron; Goffering iron.
Electric iron appeared around 1900 and became the leading term by around 1920. Steam iron is much later, around 1950. see this Ngram.
You’re looking at the term from a modern viewpoint, not a contemporary one.