Getting the Thévenin equivalent resistance for a max power problem
So the bottom line is : When and when not are voltage sources introduced in the analysis to get Rth when there’s a dependent source ?


Both should work. As a summary, you can find the \$R_{TH}\$ by doing one of the following:

Option 1: If all the sources in your circuit are independent (not in this case since there is a dependent one), you just deactivate them and find the equivalent resistance of the network you are left with. By deactivating, voltage sources become shorts, and current sources become open circuits.

Option 2: You can find the short circuit current, (\$I_{sc}\$), flowing through the terminals of interest and to find \$R_{TH}\$, you compute \$R_{TH}=\dfrac{V_{TH}}{I_{sc}}\$. In this case, you don't deactivate any of the sources.

Option 3: This is what you are doing, but you are making a mistake. You add a test source across the terminals of interest, but you have to deactivate the independent sources, and leave the dependent ones untouched. So, in your method, you need to make the 9V source a short circuit (0V).

Both option 2 and 3 should give you the right result.