I'll have to agree with the painter. We just repainted all the trim in our house and the painters used SW Pro Classic Semi-Gloss in a very white, white. It is perfect and holds up very well. I don't think it is too shiny.

You might want to check out BM Cabinet Coat, specifically designed for what you'll be doing. I've been on the paint forum many times here and it's constantly being recommended by paint experts. Here's a review of the paint.

Here is a link that might be useful: paint

Simply White is a very nice color. It was my second choice.

I don't know about semigloss for cabinets. That would be way too shiny for me. I think mine are satin finish and they have a nice luster, but no shine. My trim is semigloss and it's fine.

My trim and cabinets are different whites but there are just two places they meet, both at 5" baseboards and totally inconspicuous. My trim is way too white for cabinets.

BM Simply White is the color I chose when we repainted all the doors, trim and ceilings in the house. Very happy with it. A nice crisp white, but not stark.

I'm painting my cabinets white right now, and had a hard time choosing the white. Funny how you can stand there staring at four white paint chips taped up to your cabinets and go nuts trying to decide which one looks best! I almost went with Simply White but ended up choosing White Dove. I went with Benjamin Moore Advance in satin. The people at the store recommended either Cabinet Coat or Advance for kitchen cabinets. I'm not familiar with SW Pro. But if you use BM paint, the satin might have enough of a shine for you. I started with satin thinking I might switch to semi-gloss, but the satin is really nice and has much more shine then I expected. I would use satin for the cabinets and semi-gloss or even gloss for the trim. But I like shiny trim! Good luck! I should actually be painting my cabinets right now!