Thought Experiment - Stationary Electron and Moving Observer

Both fields electric and magnetic are unified into one electromagnetic field (See Electromagnetic tensor). Electric and magnetic fields are just description of the same beast from two different views.

This is analogical to 2D vector (1,1) having x component 1 and y component 1 in certain frame of reference (basis), but if you choose another, rotated by 45 degrees, it will have just one with length $sqrt{2}$. But it is the same vector, just viewed differently.

The fields aren’t observables, per se, since as you noticed they aren’t Lorentz invariant. What is invariant are the forces and torques imposed on charges by those fields. So as you boost from one frame to another, some of the electric field will transform to magnetic field and vice versa. But if you do a careful accounting of the forces generated by those fields in each frame, you’ll find they are identical.

The electric and magnetic fields are very similar to, respectively, the centrifugal and Coriolis forces on a rotating disc. An observer who is located outside the disc at rest WRT it detects no Coriolis force except a centrifugal one that tends to throw the particle, tossed at an initial velocity $v$, outside the plate. However, an observer located on the disc, besides a centrifugal force, detects a Coriolis effect that tends to make the path of the tossed particle curved.

Remember that the induced magnetic field in your example is very similar to the Coriolis force since both of them depend on the instantaneous velocity of the particle, and the fields vectors are both perpendicular to the curved path of the particle. Pondering over this comparison may help you to better understand how is it possible for a specific observer to detect a field, whereas it is not detectable from the standpoint of another inertial observer.

For simplicity, consider a bunch of electrons arranged in a line of constant linear charge density, $lambda$. It's easy to show that (in standard cylindrical coordinates):

$$ vec E = frac{lambda }{2pirhoepsilon_0}hat{rho} $$

and

$$ vec B = 0$$

Now have the linear charge moving at

$$vec v= -vhat z$$

then the electric field increases b/c of length contraction:

$$ vec E' = frac{gammalambda }{2pirhoepsilon_0}hat{rho} =gammavec E$$

and the current

$$vec j'= lambda'vec v = - gammalambda v hat z $$

leads to a magnetic field:

$$ vec B' = -frac{gammalambda vmu_0}{2pirho}hat{theta} $$

Note that this exactly agrees with the Lorentz transformation of EM fields:

$$ vec E'_{perp} = gamma( vec E_{perp} + vec v times vec B_{perp})= gamma vec E_{perp}$$

$$ vec B'_{perp} = gamma (vec B_{perp} - frac{vec v}{c^2} times vec E_{perp})= -gamma frac v {c^2} frac{lambda }{2pirhoepsilon_0}(hat z times hat{rho})= -frac{gamma v lambda mu_0}{2pirho} hat{theta}$$

with all field parallel to $vec v $ being $0$.