>later someone (Gibbs?) gave them the familiar vector calculus form.

It was Oliver Heaviside (https://en.wikipedia.org/wiki/Oliver_Heaviside) that rewrote Maxwell's original equations (20 of them in differential form) into the notation used today (4 of them in vector calculus form).

Here's a nice comparison: https://ddcolrs.wordpress.com/2018/01/17/maxwells-equations-...