Question: This question, unlike potential duplicates, does not concern nonclassical theories such as general relativity or the standard model.

Many sources found by a simple google search say that there is a common misconception that gravity is force. Instead, weight is a force caused by gravity. Gravity is a phenomena that causes this force to occur.

Is there any truth to this, or is it just pedantism in name conventions? In simple situations, such as on the ground level, this seems acceptable. However, considering bodies far away, using Newton's law of gravitation, it seems unavoidable that gravity is a force.

Answer: 
Raising this to the level of a misconception is unnecessarily pedantic in my opinion. If Alice were to say, "the planets are held in orbit around the sun by gravity" and Bob were to reply, "don't you mean the **force** due to gravity?" then Alice should banish him to Antarctica. Both parties knew perfectly well what she meant, so this is pedantry.

That being said, when I teach introductory courses I try to be careful with my language and refer to $F\_G = \frac{GMm}{r^2}$ as the gravitational *force*, and $g\approx 9.8$ m/s$^2$ as the free-fall (or gravitational) *acceleration*. I do this because for new students, language tends to shape their conceptual understanding of things, and I'd prefer they not say things like, "the acceleration down the ramp is two-thirds of gravity" in the same breath as, "the box is at equilibrium so the normal force is equal to gravity."
At first it is not necessary to postulate a force, just a field of accelerations to explain the movements of planets or falling apples.

But force can be related to elastic deformation, and if an object is pulled by a spring horizontally and without friction, it can be shown that $\mathbf F = m\mathbf a$.

If gravity is not considered a force, then $\mathbf F = m\mathbf a$ can not be general. A weight hanging by a spring is not accelerated, but is subjected to a force as the spring deflection shows. Without postulating a force of gravity, there is a net force upwards from the spring to the body and no acceleration.

But on the other hand, it is true that an object in free fall has no force, in the intuitive meaning of something that can be measured by a spring or load cell.

A way to escape of this conundrum is to define that a falling object is not accelerating, and the coordinates and method of calculating derivatives must change for that be true. But this is GR.