Bare in mind, that the client's corporate identity will sometimes have no alternative but to be printed in black and white, greyscale or CMYK.
My question is: I only used 2 spot colors, why does it say 4 process inks; 2 spot inks? Is this a problem?
There is at least one object in your file using CMYK inks. It could be a process color (CMYK, RGB, or Lab) set to a 0% tint, something like this where I have a rectangle set to [Black] at 0%:
If the rectangle is filled with [Paper] there are no process inks used:
You can set up a Preflight Profile that will flag all objects using process inks. Like this
Also, before you preflight you should remove any process colors from the document. From the Swatches panel choose Add Unnamed Colors, and then delete all the swatches that don't have a spot color icon. If any of the process swatches are in use you'll be asked to choose a replacement and you can replace with one of your spots.