Why do I have to consider the controlled voltage generator?
Think of it this way, if you connect a test source across A and B terminals, the test source cannot affect the current through the independent source - that's why it's called an independent source - it's value does not depend on the attached circuit in any way.
However, the voltage across E1 will, in general, be affected by the test source and, thus, the equivalent resistance seen by the test source is modified by the presence of the dependent source.
And thereby why I cannot eliminate the current source I1 and in the
example in question it gets replaced by an 1A current source?
If the 1A current source mentioned is, in fact, the test source, you should zero the 5A source to find the Thevenin resistance of the circuit.
With the 5A source activated, there will be an open circuit voltage, \$V_{AB_{(OC)}}\$. When you connect the test source, the voltage \$V_{AB}\$ will be different from the open circuit voltage. To find the Thevenin resistance, take the difference in the voltages and divide by the test source current.
But, you get the same result if you simply zero the 5A source which sets the open circuit voltage to zero. Then, you get the Thevenin resistance directly from the value of voltage across the test source.
In other cases it is true to assert that the controlled generators are
not to be eliminated and so the other generators (current and voltage)
are to be replaced by unitary generators?
I honestly don't know where this idea comes from. A unitary generator is typically used as a test source but I'm not aware of any reason to replace the other sources. Perhaps you should expand this question a bit. I suspect there's a misunderstanding here.
SUBSTITUTION THEOREM:
If the voltage across and the current through any branch of a dc bilateral network are known, this branch can be replaced by any combination of elements that will maintain the same voltage across and current through the chosen branch.
As long as terminal voltage and current is same, accordance with substitution theorem, you can substitute whatever in the branch. Here is an example that demonstrate how it works.
THÉVENIN’S THEOREM:
Any two-terminal dc network can be replaced by an equivalent circuit consisting solely of a voltage source and a series resistor.
You have asked an important question indeed. Thévenin’s equivalent circuit has a series resistor but in the second circuit diagram I have only used a source and in the third I have used both (more possible). Both are accordance with substitution theorem.It means one can replace a branch with any combination of elements which is not true for Thévenin’s theorem.For the marked branch in the main circuit if you use Thévenin’s theorem you will get Vth or Eth = 0V and Rth =3 Ohm. This is because Thévenin’s theorem doesn’t care about rest of the network or the load resistance but substitution theorem does. Without the whole circuit substitution theorem is not applicable but in Thévenin equivalent circuit the load resistance may vary.
Best Answer
You are correct. R4 must be taken into account when calculating the Thevenin resistance. The Thevenin resistance is defined as the resistance looking into the circuit from the load terminals. It is clear that R4 is in series with the rest of the circuit. Imagine if R4 was infinite. According to your classmate, it would have no effect on the Thevenin resistance when it is clear that it makes the resistance infinite. Don't rely on your classmates but use the knowledge that you have gained in the classroom. That is why you are there.