Why white ? I do agree if white is the paln all the same color just different sheens makes this much easier.I do not see the backsplash but I would not let it decide color for walls and cabinets . Waht is the reason for painting the cabinets at all? Take pics of the space not you and then maybe you will get some help. We need to see the backsplash and take a pic with the Greek Vills color card next to it. I think whites will be tricky with the floor in the kitchen.

I would not paint the cabinets, they are already very nice, and since wood is back in style, you are already on trend. I would paint the walls a color rather than white, but, if you really want white, then, use the same white just in different sheens. Eggshell or matte finish on the walls, flat on ceilings, and your choice of semi gloss or satin on the trim, depending on how shiny you want your trim.

How about Shoji White with Pure White? Or Greek Villa throughout. I also wouldn't go to the trouble of painting the kitchen cabinets unless the existing finish is worn or damaged and can't be refreshed.

What are you trying to accomplish?

Your space looks great in the photo and the cabinet stain color seems to work well with your wall color and floors. Your backsplash is a little cool but still looks good.

I thought I was set on greek villa but then I read don’t put it next to pure white

Whoever told you that doesn't have one fat clue what they're talking about.

Greek Villa and Pure White are an ideal white-on-white pairing. Because:

1. They're similar in hue family, they sit near each other in color space in the 102 to 107 range. They are literally related to each other in terms of hue. Color harmony is about threads of relationship between/among colors. Hue is the easiest dimension of color to thread. E.g. Greek Villa and Pure White.

2. In addition to the hue family thread of relationship, they're also similar in Value - another color harmony thread.

3. Most significant is there's a generous difference in Chroma. That means contrast so you can tell they are two different colors and neither will make the other look dirty of dingy.

There is no logical, reasonable or technical reason not to use the two together. If you like it, think they look pretty together, then do it.