I need to know how LightBurn saves layer settings

It depends on your preferences / settings.

There are a couple things worth mentioning here:

Units affect how settings are displayed, but don’t actually change the settings themselves. For example, if you change from mm to inches, it’s possible for the speed settings to change as well. 200 mm/sec could be displayed as 7.9 inches/sec. There is a “mixed mode” setting that only changes distance units, not speeds.

Default layer settings, if enabled, are per-laser, so if you have more than one laser configured, and have set layer defaults for each, the values for a new layer will come from the current laser profile.

If you have never specified any defaults, and have “Load default layer settings” enabled, you’ll get LightBurn’s built-in defaults per layer (100mm/sec, 20% power)

If you have that toggle switch disabled, layers are just “the last thing you set”. This can be either a blessing or a curse, depending on how you work. If you load someone else’s project file, it’ll overwrite your layer values, and stay that way until you change them back.

Be sure you Open project files, not Import them. If you import a file (File → Import, or the Import button on the toolbar), it pulls in only the graphics, not settings. If you open a file (File → Open) then it reads the file along with all layer settings.