I understand what you are saying but what I am not understanding is how the current in wye connected conductors in a balanced system (no neutral current) could be different from a delta connected system where the line current is the same. It's the same amount of current through three wires. Don't just tell me they are different; explain how/why they are different. I know a lot but I don't know everything.

I personally think the common nomenclature "phase current", "phase conductor", etc. may be adding confusion because each line output conductor also has an associated phase relationship with the other two.

Be that as it may, as Ethan stated the phase current is the current flowing through each individual 2-terminal winding (or inverter). In a delta two of the three windings are connected to form each line output, and therefore because of Kirchoff's law each line output current is the sum of the currents on these two connected windings. On a delta, a 180 degree reversal (polarity flip) is effectively made on successive phase windings so the currents are 60 degrees apart from each other (instead of 120 as in a wye) in order to make the resulting vectors form a triangle (i.e. delta). Because the two phase currents are 60 degrees apart from each other the resulting line current has a phase right in between these two currents, i.e. +-30 degrees from each one. Therefore the component of each phase current that contributes to the line current is scaled by cos(30) = 0.866 because that's how much is "in phase" with the resulting line current. And so total line current is 2 x 0.866 = 1.732 times the current in one of the windings, instead of 2 times if the two windings were in phase with each other.

So back to the orginal question (I think): Consider a wye and a delta connected device (transformers or inverters) that both produce the same line currents and L-L voltage on the three output wires, and with a balanced load. How do the currents compare on the "phase conductors" that that are connected to make the wye configuration vs. the delta?

On a wye, only one phase conductor is connected to each line output and so the line current is the same as the phase current. On a delta, two phase conductors are connected to each line conductor and therefore (as above) each contributes 0.866 times its phase current, for a total line current of 1.732 times one of the phase currents.

However on a wye, because the voltage across each phase winding is 120 degrees apart with a common connection at the neutral, the L-L voltage is the vector sum of two phase voltages and therefore 1.732 times the phase voltage. On a delta the L-L voltage is of course the same as the phase voltage.

So in order to make the wye and delta output have the same line output voltages and currents using the same transformer primary circuit, the delta windings would need to have 1.732 more turns than the wye windings (so they produce 1.732 times the voltage and 1/1.732 times the current as the wye ones). This would result in the same total output power for the same balanced load.