Question about air sharing

Would you share the main stage, or the secondary stage?

I just got my open water certification from SSI, and I think I remember my manual saying you should share the main stage, and use the secondary yourself - also called the octo, I think? But when I dived with my instructor, she taught me to share the secondary.

My instructor is certified under both SSI and PADI (maybe others), and when I looked it up I did see some people mention that sharing the secondary/octo is the PADI way, and sharing the main is the SSI way. Is that true?

Which way were you taught? Which would you do? If you're an instructor, which way do you teach? And more importantly, why?

To me, how my instructor taught me makes the most sense, because what you hear in most rescue situations is that you have to make sure you are safe first so you can be in a position to help; so it follows that you should keep breathing out of your main and share the secondary (instead of interrupting your breathing to switch).