I can sympathise with your situation; for reasons unknown to me, people are taught in ways that often leaves gaps where such questions/misconceptions originate.
In general, to avoid confusion it is best to define anode and cathode in terms of reduction/oxidation. The anode is the electrode where oxidation takes place, and the cathode is the one where reduction takes place.
Also, in an electrochemical cell, electrons don't flow, but ions do.
Now, let's address the differences between galvanic and electrolytic cells.
In a galvanic cell the reaction proceeds without an external potential helping it along. The oxidation reaction which produces electrons takes place at the anode, results in a build-up of negative charge at the anode in the course of a reaction till equilibrium is reached. Thus the anode is negative.
On the other hand, reduction takes place at the cathode, and consumes electrons, and thus leads to a build-up of positive charge (or depletion of negative charge). Thus the cathode is positive.
The situation is reversed in an electrolytic cell where you do provide an external potential, and you are essential pushing the reaction in the reverse direction (say, when you charge a cell). So while oxidation still takes place at the anode, the anode is now positive and while the cathode is still the site of reduction, it is now negative.
You can read more here
In physics, circuits usually depict a current which moves from the positive electrode of a batter to the negative one. This is not representative of the flow of electrons; it is a historical artifact of sorts, because this convention was agreed upon before the discovery of electrons. Read more here