Solar panels for a DC solar pool pump
First of all, are those numbers for the pump all from the same model? Because the range of the numbers is rather large: you say that it starts with 2 watts, nominally runs at 15 watts(?), might sink 4.5 amps at 12 volts, which gives 54 watts. And even larger numbers are calculated in your question, which probably aren't relevant. 

Looks to me that even the 120 watt panel is overkill, but it can be connected directly to that pump if the pump can inded handle 24 volts, which is more than the open circuit voltage, which is the maximum voltage the panel can ever generate.

The thing to realize here is that it's the amps that turn the motor, not the volts. Volts are something that have the capability of giving amps to a load. And pushing amps through a load generates volts across the load. In your case, there's a solar panel, which is more like a constant current source, unlike, say, a battery, which is more like a constant voltage source. 

When you connect the solar to the motor and there is sun, the panel pushes amperes to the load. The 120W panel looks like it might push something like 6 amps to the motor. That would perhaps generate something like 16 volts over the motor, although this number is only based on a rough feeling. That would mean that some 96 watts are dissipated somewhere in the pump - and that seems quite a lot for a pump that nominally takes 15 watts. I don't know; maybe the pump is electronically regulated somehow so that it never draws so many amps...?

All in all, I have a bit uneasy feeling about this. It would help if you could provide some more info about the pump. A link to a datasheet or a product page, maybe?