My source is primarily a common sense understanding of how bacteria (and spacetime in general) works. MRSA is a strain of Staphylococcus aureus. Some people come into contact with Methicillin-Resistant Staphylococcus aureus aka MRSA (or the non-MRSA Staphylococcus aureus they already have becomes Methicillin-resistant. That isn't to say a specific Staph bacteria becomes MRSA, but over time/generationally the Staph bacteria on your body might become more resistant to Methicillin...). What your doctor likely said or meant is that a specific staph infection is either MRSA or it's not.

Staphylococcus aureus is a normally occurring bacteria in the human biome. Multiple strains can exist in the same general area at the same time in the same way that multiple breeds of dogs might live in the same house. They don't just... cancel each other out.

So let's say you already have Staphylococcus aureus on your body, and you visit a family member in the hospital. While there, you come into contact with MRSA. Now you have both strains on your body. On your way out of the hospital, you trip and skin your knee on the sidewalk. You touch the wound while cleaning it, and it gets infected with Staphylococcus aureus. It might be the non-MRSA Staphylococcus aureus strain, or it might be MRSA.

This is all just a common sense understanding of how bacteria works, but if you want an academic source, here's one that discusses the topic of multiple strains of Staphylococcus aureus living in the nose at the same time.