This doesn't really help with programming, but in physics it's traditional to use up- and down-stairs indices, which makes the distinction you want very clear.

If input x has components xⁿ, and output f(x) components fᵐ, then the Jacobian is ∂ₙfᵐ which has one index upstairs and one downstairs. The derivative has a downstairs index... because x is in the denominator of d/dx, roughly? If x had units seconds, then d/dx has units per second.

Whereas if g(x) is a number, the gradient is ∂ₙg, and the Hessian is ∂ₙ∂ₙ₂g with two downstairs indices. You might call this a (0,2) tensor, while the Jaconian is (1,1). Most of the matrices in ordinary linear algebra are (1,1) tensors.