This seems broader than the other question which was interpreted mainly in the “category theory” sense (1.).
An early, maybe earliest, case of the “normal form” sense (4.) is Jacobi (1837) calling canonical the “Hamilton form”a of the equations of mechanics, and also any variables, coordinates or “elements” in which they take this form; today we would speak of Darboux coordinates or Darboux normal formb of a symplectic structure. Remarks:
A big difference with the case of fractions is that the normal coordinates (or isomorphism to normal form) are far from unique.
Similar: the Jordan canonical formc and Frobenius rational canonical formd of a matrix.
So far as I can tell, Jacobi may well have originated the phrase “normal form” too (1845, 1850).
a notoriously used before by Lagrange and Poisson.
b called canonical by Frobenius (1877), canonical or normal by Darboux (1882).
c called canonical by Jordan (1870, p. 114); Kronecker (1874) bitterly deplored the terminology.
d called normal by Frobenius (1879, pp. 207-208).