Why are the four fundamental interactions still called "forces" if force is not fundamental in modern formulations of physics?

In introductory physics, force is presented as a fundamental concept through Newton's second law, (F = ma). However, as one moves to more advanced formulations, force seems to lose its fundamental status.

For example:

• In Lagrangian mechanics, the equations of motion are derived from the principle of stationary action.

• In Hamiltonian mechanics, the dynamics are expressed in terms of the Hamiltonian and canonical equations.

• In General Relativity, gravity is described by spacetime curvature rather than as a force.

• In Quantum Field Theory, the fundamental objects appear to be fields, gauge symmetries, and interaction terms in the Lagrangian, rather than forces.

Given this, why does modern physics continue to refer to the four fundamental forces instead of the four fundamental interactions?

"force as the cause of motion" to a "tree-god as the cause of growth,"
"When we say force is the cause of motion, we talk metaphysics, and this definition, if one were content with it, would be absolutely sterile. For a definition to be of any use, it must teach us to measure force; moreover that suffices; it is not at all necessary that it teach us what force is in itself, nor whether it is the cause or the effect of motion." — Henri Poincaré

More specifically:

1. Does the word force have a precise mathematical definition that applies uniformly across these theories, or is it primarily historical terminology?

2. In what sense are the strong, weak, electromagnetic, and gravitational interactions "forces" in modern theoretical physics?

3. Is there a rigorous mathematical criterion that distinguishes a force from a more general interaction?