On the idea of an inertial frame of reference and its theoretical interest An inertial frame is often defined as
(1) a frame that is not accelerated
(2) a member of an (equivalence) class of frames , that is , the class of frames that are in constant rectilinear motion with respect to one another
(3) a frame in which newton's first law holds.
I've tried to understand why this idea of an inertial frame is so important in classical physics.
What I arrived at is that :
(4) an inertial frame is one in which any acceleration is caused by a true force, hence a true " cause"  ( not a pseudo-force such as inertial force, centripetal force, coriolis force etc) , that is by an effective interaction between two bodies
(5) in all inertial frames, the same forces ( hence " causes" of acceleration) are observed, because though bodies may have different position vectors and different velocities in these frames, they   have the same acceleration vector ( at a given time).
So it seems to me that considering inertial frames allows physics to observe the true facts ( objective accelertions that are the same "everywhere") and to look for the true causes ( true forces ) which again are the same in all these frames.
Do the above statements capture the theoretical interest of " inertial frames".
If what I said is correct, why has physics felt the need for laws that hold for all frames of reference (not only inertial frames)? Is this need the motivation behind einsteinian relativity theory?
 A: I think your conclusions make sense. I will just try to put it in a different way for clarity.
In a mathematical point of view, we are looking for a transformation which preserves the form of the laws, which are just mathematical equations. A consequence of that result is that all the observers in the frames which are in constant rectilinear motion with respect would report the same form of equations in their experiments performed. This can be seen easily in terms of Newton's first law as position appears in the equation with a double time derivative; and any change to the position in the first order will not make a difference.
Though, I do not think I agree with your last question. The theory of relativity is motivated by the need that physical laws need to be applicable in every reference frame. It was rather the experimental evidence which mounted up at the end of 19th and beginning of 20th century. Most importantly, the Michelson-Morley experiment. The debate on Ether, the medium  for light, was what motivated Einstein to come up with relativity in my understanding.
A: The concept of inertial coordinate system goes to the phenomenon of Inertia.
Inertia: a force is required in order to cause change of velocity.
(There is currently no theory that attempts to explain inertia: in order to formulate a theory of motion the properties of inertia must be granted.)
Inertia is never not there. We can construct setups such that friction is completely eliminated, but there is no avoiding being subject to inertia.

How Inertia gives rise to an equivalence class.
Before turning to actual Inertia I start with the following thought demonstration. You are on a ship on a large body of water. The water itself is, of course, identical everywhere, the water is perfectly featureless. So you cannot establish a reference of position.
Initially the ship does not have a velocity with respect to the water. If you start on a journey, can you return to your exact starting point?
As we know, even though you cannot establish a reference of position, you can return to your starting point, by using dead reckoning. During the entire yourney you can measure your velocity with respect to the water, you then use integration to keep track of your position with respect to your starting point.
Let there be two ships, they go on separate journeys, planning to rejoin at a particular point in space and time. With perfect dead reckoning this plan is achievable. The better the accuracy of the dead reckoning, the further ahead in time the point of rejoining can be.
The property that allows the dead reckoning to be such a powerful instrument is the same as the property that prevents you from having an intrinsic reference of position: the water is featureless. No part of the water has a velocity with respect to the rest of the water. It is that very uniformity of the water that provides you with a global reference. While you don't have a reference for absolute position, you do have a global reference for your current position with respect to your starting position.

In space
Moving on to inertia we set the thought demonstration in space, sufficiently far away from gravitating bodies such that gravitational effects are negligable.
So the starting condition is a spaceship, floating in space. If you start on a journey, can you return to your exact starting point?
Obviously there is no such thing as measuring your velocity with respect to space. Space does not have any identifyable part that you can assign a velocity vector to.
In theory of motion we have these three tiers:
Position
Velocity
Acceleration
The phenomenon of Inertia allows you to keep track of your journey, but the thing that is measured is moved up a tier.
You continuously measure the direction and magnitude of your acceleration, and to keep track you perform two tiers of integration of those acceleration readings. You integrate the acceleration readings to velocity values (with respect to your starting point), and then you integrate those velocity values to position values (with respect to your starting point).
Two spaceships can go on separate journeys, and they can rejoin at some point in the future. The better the accuracy of the inertia dead reckoning, the further ahead in time the point of rejoining can be.
Analogous to the case of dead reckoning at sea this is dependent on Inertia being uniform throughout space.

The equivalence of the equivalence class of inertial coordinates systems expresses that inertia is universally uniform.
Conversely, imagine a state of affairs where inertia is not uniform throughout space. Then two ships cannot use inertia dead reckoning to rejoin at some point in the future.

There is a psychological dimension that needs to be understood. The human brain tends to overlook things that are the same everywhere. Example: if you wear goggles with tinted glass then after a while you stop noticing that the glass is tinted. When something is everywhere your brain starts finding ways to not be aware of it.
(The reason I mention this psychological aspect: there is a trend in physics to not mention the phenomenon 'inertia' anymore, instead circumscribing things in terms of 'inertial frames of reference'. The result is loss of clarity.)

In any viable theory of motion inertia is the prime organizing principle.
The concept of equivalence class of inertial coordinate systems expresses that prime organizing principle.

As we know, there was the shift from newtonian dynamics to relativistic dynamics.
In terms of newtonian dynamics the relation between any two members of the overall equivalence class of inertial coordinate systems is expressed by Galilean transformation.
In terms of relativistic dynamics the relation between any two members of the overall equivalence class of inertial coordinate systems is expressed by Lorentz transformation.

That is, the concept of equivalence class of inertial coordinate systems carried over from newtonian dynamics to relativistic dynamics. The shift was in the concept of how the members of the equivalence class relate to each other.
