You can’t specify a depth with a laser. You can test and get an approximation for that particular piece of material, but it’s never going to be perfect every time. That’s the realm of a CNC router, not a laser.
Create a series of circles and run them at different speeds and power and you can measure the depth. Find one you like and mark the settings used. It may well be different with a different piece of wood, but it will probably be close.
When engraving you are burning a area with the same power, but even the annual grain will burn easier and not, so you will have visible raised grain on the bottom. You can sand it more even or just go deep enough not to matter or cut all the way through and put it together like a puzzle. There are many threads , videos, etc to provide a very deep dive into that rabbit hole.
It is difficult to do an inlay in solid wood. The density and grain structure messes with getting a clean and even pocket depth.
The MDF core of Proofgrade plywood is very good for an inlay. You can get an even depth of pocket.
There are different power level, speeds and LPIs for achieving the effect you want. You might be tempted to just burninate it all at once. That can leave a buildup of soot at the edges of the engrave. I often do less aggressive engraves and do multiple passes. That way the burn gets blown away rather than accumulating around the edge of the inlay.
I also clean the edges of the inlay with a dental pick to get the corners of depth sharp.