All the arguing about arrow directions and conventions is like arguing whether buses should be painted red or green. The truth is that whatever direction voltage and current arrows point doesn't matter as long as you define your convention and apply it properly during the analysis. Similarly, you don't have to draw loops on the diagram if you are doing a current balance around nodes, you just need a direction arrow to give a sign convention for each current path. You only need the loops if doing a mesh analysis summing voltages around a current loop.
This is all part of the "deeper" understanding of concepts and theory, which goes beyond simple mechanical application of formulas.
Ultimately there is no single "correct" solution to a problem like this. There are many equivalent solutions that may be obtained depending on where you place your reference point for voltages and which direction you draw your current arrows. There are as many correct solutions as there are permutations of these choices.
It's quite true that the choice of branch current assumed directions is arbitrary, and will have no effect on the final result. Likewise, any node can be chosen as the reference node for a nodal solution, and won't affect the
relative values of node voltages in the result solution.
But this ability to make arbitrary choices about assumed directions and polarity of currents and voltages does not apply to the 3 independent sources. The two possible ways to interpret the polarity of those sources give results with differing signs even if consistently applied to all 3 sources at the same time. Circuit voltages are always measured with respect to a reference. Considering only V1, if we choose the bottom of the source as a reference then the top can be either 120 volts or -120 volts. It has both a magnitude and polarity. Without the arrow how would we know the polarity? The arrow tells us the polarity, and the polarity matters, so the choice of whether the polarity indicates rising potential or falling potential matters.
As far as whether there are many correct solutions, I think this is a question of semantics.
Different choices of reference will result in different values of node voltages with respect to that reference, but the constellation of node voltages all have the same values relative to one another. There is only one correct constellation of relative node voltages.
The same sort of reasoning applies to branch currents. If one person assumes the current in Z5 is upward, and another person assumes it's downward, the two solutions will result in the same magnitude for that current but with different signs. After reconciling the sign of the calculated current with the assumed direction, the actual direction, up or down, can be determined. The "actual" directions of branch currents are unique, and there is only one correct solution for actual branch current directions.
The original problem only asked for a current I, which is the current out of the bottom of Z4. There is no single "correct"
method of solution, but there is most assuredly one, and only one, answer for the current I which is correct in both sign and magnitude.
The direction of the required current I is indicated on the diagram, and that direction is not subject to arbitrary choice; it's part of the problem description.
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