You are not making changes to any specific layer, you are making changes to the underlying data which is common to all maps generated from OSM data (some maps will use additional 3rd party data, but that is a bit out of scope here). Now why there is a difference in the display of what you changed between CycleOSM and the standard style can be due to a lot of things, from different choices on which data to render, to simple caching issues in your browser.browser. As a rule the "standard" layer is the quickest to update, so I suspect the issue is due to rendering differences, but without more information you are not going to get a definitive answer.

You are not making changes to any specific layer, you are making changes to the underlying data which is common to all maps generated from OSM data (some maps will use additional 3rd party data, but that is a bit out of scope here). Now why there is a difference in the display of what you changed between CycleOSM and the standard style can be due to a lot of things, from different choices on which data to render, to simple caching issues in your browser.