Why does heat flow from a region of higher temperature to one of lower temperature?

and so the direction of heat flow was just a matter of definition.

It is not just a matter of definition. It's a matter of observation. Heat is never observed to flow naturally or spontaneously from a low temperature substance to a high temperature substance. For that to happen, work must be done (as, for example, in the case of refrigerators and heat pumps.)

Before the second law was developed we had the first law. That law is a statement of conservation of energy. If heat heat were to flow from a cold to hot object, the heat lost by the cold object would equal the heat gained by the hot object and the first law would be satisfied. But since this never happens, a new law was needed that made such a flow of heat impossible .

Although the wording of the "argument" is rather awkward, at least to me, it is true that when heat transfers from the hotter system there is a decrease in both the entropy and internal energy (dU) of the hot system and there is an increase in both the entropy and internal energy of the cold system. The important distinction is internal energy is conserved but entropy is not, unless the heat transfer is "reversible".

The decrease in internal energy of the hot system exactly equals the increase in internal energy of the cold system, per the first law, for a total change in internal energy of zero. On the other hand, for any finite difference in temperature between the hot and cold system, the decrease in entropy of the hot system is less than the increase in entropy of the cold system, for a total entropy change greater than zero.

If the hot and cold systems are thermal reservoirs (constant temperature sources and receivers of heat), the entropy change of the hot system is

$$Delta S_{H}=-frac{Q}{T_H}$$

The entropy change of the cold system is

$$Delta S_{C}=+frac{Q}{T_L}$$

The total entropy change is

$$Delta S_{tot}=+frac{Q}{T_L}-frac{Q}{T_H}$$

You will note that for all $T_{H}>T_{L}$, $Delta S_{tot}>0$.

So, is the direction of heat flow because of definition or the 2nd law of thermodynamics?

No. The natural direction of heat flow from hot to cold is an observable fact of nature. The second does not dictate nature. The second law reflects nature. The second law says that

$$Delta S_{tot}=Delta S_{sys}+Delta S_{sur}ge0$$

Where the equal sign applies to a reversible transfer of heat, i.e., when the difference between $T_H$ and $T_L$ approaches zero.

If heat flowed from cold to hot, the signs for the entropy changes in the first two equations above would be reversed, resulting in $Delta S_{tot}<0$ in violation of the second law.

Hope this helps.

Heat is, among others, kinetic energy of the disordered microscopic motion of atoms and molecules. You probably know that kinetic energy is exchanged during elastic collisions between two masses (think of pool billiards). It is the same for atoms and molecules: since there are so many of them, they are constantly and frequently bumping into each other, exchanging kinetic energy. A hot object is characterized by fast moving atoms/molecules, a cold object is characterized by slow moving atoms/molecules.

Now think about what happens if you put something hot close to something cold. The molecules inside the hot object will bump into each other just like before and nothing much will change there. The same inside the cold object. But at the interface between the hot object and the cold object, fast molecules from the hot object will collide with slow molecules from the cold object. To the effect that the formerly slow molecules become faster and the formerly fast molecules become slower. Of course, this also happens in reverse, which results in averaging out the differences between both objects.

Since the molecules near the interface stay in contact with molecules from the insides, this kinetic energy transfer progresses right towards the cores of the respective objects. Therefore, heat flows from the hot to the cold.

Since these motions are disordered, you cannot avoid that this is happening, just because you do not know what is happening in detail. You can't just turn one object by an angle of 10 degrees and hope that there will be less collisions. Nor can you put your finger on an individual molecule and keep it from bumping into another one. No matter what you do, disorder is increasing. Before, there was a tiny little remaining amount of order in that you knew that one object's molecules were faster than the molecules in the other object. Afterwards, both objects eventually have the same temperature, which means order has decreased (you know less about the molecules than before). This is the essence of the 2nd law of thermodynamics: disorder (aka entropy, aka negative information) will always increase without your consent.

The zeroth law basically states that systems being at equilibrium with each other is a transitive property. if A is at equilibrium with B, and B is at equilibrium with C, then A is at equilibrium with C. From this we can map equivalence sets of systems at equilibrium onto the real numbers. If you do it in a particular way, you get "temperature."

However, this says nothing about how heat transfers between systems tagged with a larger temperature and systems tagged with a smaller temperature. That gets done with the second law.

And also, its worth remembering that when we see a "zeroth law," it usually means that the first few laws (like the 1st, 2nd, and 3rd) came first, and only later did we realize that we needed something more fundamental.

Suppose we mix two gases that initially have different temperatures. Higher temperature gas molecules move faster than lower temperature gas molecules. Collisions occur between the molecules of one gas and the other. Part of the energy from the faster molecules passes to the slower molecules. Gradually the temperature of both gases equalize. We have called Entropy to the natural tendency of the spread of energy , and we have given it the rank of law, it is the 2nd law of thermodynamics. Each time the energy goes from a smaller to a bigger space ( a bigger amount of matter or the same matter in a bigger space), the entropy increases. The quantification of this dispersion is the entropy value. Clausius expressed the 2nd law by saying that heat always goes from high to low temperature. However, Boltzman introduced a probabilistic characteristic to the 2nd law, he came to tell us that if the heat goes from high to low temperature it is not because there is a universal law that orders it so but because there is an overwhelmingly greater probability that it is so make it the opposite. Nature has a tendency towards equilibrium, when the balance is reached, the energy is no longer useful, that is why the Carnot machine needs both a hot and a cold focus, which is another expression of the 2nd law. The transfer of energy through heat flowing from hot to cold matter causes the useful energy to become energy from the particles in the environment from where nothing can extract it, useless energy. Every time this happens the entropy of the universe increases. When the universe reaches thermodynamic equilibrium it will have reached its maximum entropy and nothing will happen in it anymore. Obviously it will happen in a long long time.