How Do I Stop My Old Browser From Opening When My New Browser Is Set To Default?

I have installed a replacement browser because I don’t like Safari. When I was using Safari, I told certain sites to notify me. When I click on the banners and/or alerts, Safari opens DESPITE THAT I’VE MADE THE NEW BROWSER THE DEFAULT BROWSER and despite that the new browser is already open. Question: How do I stop Safari from opening when I click on my banners and alerts? Reminder: The new browser is set as the default browser. Please don’t tell me all the good things about Safari and that I should be using it. I would just greatly appreciate an answer to my question. Thank you.
I don’t think it’s right that two browsers are open simultaneously. Waiting for Safari to open while Firefox is already open is a waste of my time.
—–
Gail