Can I say:
It opens the browser on the URL [X]
meaning that something is opening the browser with the URL [X] already typed in and loading?
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Yes, you may. With or at are more common, but a Google search for "browser on url" shows plenty of hits with this usage, for example: How to open a browser on a given URL
You may open url in/with a browser.
Order of the action: usually you open a browser, then the browser opens url.
In case you have some "shorcuts" on your "desktop" you may open "the browser with the url" by clicking it. That means, the browser will be opened with the url adress embedded in shortcut.
You may "open url in a browser" - you type the url to the browser address bar.
Personally, I have never heard of open browser on url.