So $m\vec a$ is not a force. There is a subtle difference.
A force is defined as a disposition to accelerate, not an acceleration itself. This is a fancy word that just means “if nothing else intervenes, then the thing accelerates.” Furthermore it is indeed weighted by this amount of stuff parameter $m$ which says “if the same disposition is applied to twice the stuff, and nothing else intervenes, then the thing accelerates half as much.”
But for example sometimes things are in a state of force balance. Gravity is pulling down on me, but my chair is pushing up on me, and as a result I am not accelerating. But the forces are still there. Gravity still disposes me to want to go downward and the force of the chair still disposes me to want to go upward, it’s just that they happen to be perfectly balanced against each other so that I do neither. (I must pause to say that this has nothing to do directly with Newton’s third law. Many newcomers make the mistake of confusing force-balance with the third law, they are also different.)
One really clever thing about this definition of forces is that, if you imagine someone moving past me on a train, maybe they are juggling balls on the train, I see all of these balls move past me with some great velocity of around 90 km/hr or however fast the train is going: but we both agree on any change in velocity of the balls, and our clocks both agree on how long a second is (until we get to relativity but let’s ignore that), so we both agree on any accelerations we see, so we both agree on the forces we see on the balls, even though we disagree about the actual momentum and energy that the balls are carrying. Very handy!
And like sometimes forces are not balanced, and you see an acceleration in some direction. There might not be any forces actually pointed in that direction! So for example sailboats can sail into the wind by cleverly coordinating both the wind (which obviously pushes away from the wind) with a sail and a rudder and a mainboard anchored in the water, which each pushes in some completely other direction. You sum up these wildly different forces in wildly different directions and you get a "net force", and a resultant acceleration, in a totally different direction from any of them. I cannot say that such an $m \vec a$ is a force because there is no force pointed in the appropriate direction. But it is the vector-sum of forces on the boat.
Are there types of forces?
Now beyond this you are asking whether there are substantially different types of forces, and the answer is “not at the level that you would like: but there are substantially different reasons we care about forces, and we label these forces by the reasons we care about them.” So for example the thing you are calling the “normal force” is caused by the fact that my chair (and the floor beneath it and the planet beneath that) has not yet broken, so I cannot fall through it. It is what we call a constraint force, it is created by a constraint on the motions of the system. We call it “the normal force” so that you know that it is the force that is normal (fancy old word meaning “orthogonal to” or “perpendicular to”) to the constraints of the system. But it is ultimately caused by electromagnetic and Pauli repulsion between electron clouds of atoms, so if we were classifying it by type we would have called it a “quantum force of Pauli repulsion between electron clouds because electrons cannot be in the same state as each other”. But we don’t classify it that way, we classify it by some sense of why we care about it, what it is doing in our equations: it is stopping me from falling through the floor, it is enforcing a constraint that I do not think is going to happen in the physics of this system. Similarly a tension force is expressing that there is a stretchy thing that has been stretched out past its equilibrium length and it wants to return back to its equilibrium length: that’s not the underlying type which is again an “electromagnetic force of atoms pulling on nearby atoms, plus probably an entropic force of long stringy molecules being forced to straighten up but thermally they really want to go back to being in complicated squiggles”. But we call it “tension” because we don’t care why the rope or spring wants to return back to its equilibrium length, it is sufficient that it does want to, and that is the reason why we care about it.