Sorry, you've triggered my inner Word Nerd...!
Pinny is short for pinafore, but only in the original sense, meaning either an apron fastened for protection over a skirt or dress, especially one that includes a bib, or a child's sleeveless over-garment, used for the same purpose and tied at the back. I was certainly familiar with the word "pinny" in my youth, though I am not old enough to remember a time when, according to the Oxford English Dictionary, "children's pinafores were originally worn by both boys and girls". (It is quite literally a "pin-afore", a garment that is "pinned on the front of" a dress or, in the case of a boy, perhaps a trouser suit, a sailor suit, or, if very young, a dress.)
A pinafore dress (colloquially "pinafore" for short) is a dress designed in a similar style to a traditional pinafore, i.e. a sleeveless, collarless dress worn over a separate top (blouse or jumper=sweater); but I have never heard such as dress called a "pinny".