About magnetizing inductance
Magnetizing inductance is not equal to mutual inductance.

It is for a 1:1 transformer. The mutual inductance (the turns that are 100% coupled between primary and secondary) represents the magnetization inductance. Those turns that don't couple are not mutual inductance but leakage inductance. Mutual inductance is \$\sqrt{L_P\cdot L_S}\$ hence, for a 1:1 transformer where primary and secondary inductance = L, M = \$\sqrt{L^2}\$ = L.

Mutual inductance refers to the energy which is transfered from the
primary to the secondary.

The presence of mutual inductance allows energy to pass from primary to secondary.

The energy which is not transfered from the primary to the secondary
refers to the leakage inductances

Leakage inductance is that inductance that doesn't couple primary and secondary

Magnetizing inductance correspond to an inductance which is placed in
parallel to the primary side of the transformer.

Correct, for the model of a transformer where &quot;the transformer&quot; is internally an ideal power transfer component.

So the energy stored in the magnetizing inductance (1/2LI²) is not
transfered to the secondary.

Correct (except for a flyback converter), but the magnetization inductance also facilitates primary to secondary power and energy transfer. It does two jobs.

Apparently in a forward transformer, if the magnetizing inductance was
infinite, the flux into the transformer would be equal to 0

Not really a practical issue.

the flux at the primary would be cancelled by the secondary.

The flux due to load currents in the secondary are cancelled by flux in the primary due to those same load currents whatever the inductance.