Show the picture to your painter. Yes it's a glaze. There may be faux distressing too, can't really tell.

The cabinets look like frameless overlay cabinets with raised panel doors and some kind of applied molding. Take a look at a cabinet catalog and see if you can reconstruct it, or show to your cabinet maker.

The cabinets were done by the company in the link. This finish would be difficult to achieve by a custom cabinet maker (IMO). The Houzz link states that it is a proprietary, factory finish.

Ask at the kitchen forum.

Here is a link that might be useful: Link

google 'antique glazed cabinets' there is a lot of info.

There is an art to it so getting an exact match is more than just knowing what products where used.

If you are dealing with a cabinet maker that doesn't understand how to produce the desired look, then you have bigger problems than trying to explain a glazed finish to them. Any custom shop should be able to provide you with several samples from which to choose, and every single cabinet line out there has multiple choices as well.