Component copy vs insert (I think) (actually paste vs paste new)

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OK - what I'm describing is all within a single design, no use of external components.
I created a component within this design and have copied it so that I now have 13 (copies? instances? not sure what term to use). So what now happens is anytime I edit or modify one of the copies all the other copies get the same modifications. Essentially the components are linked although that may not be the right term as "linked" components are indicated with a little chain-link and these components don't have the link icon.
So how do I un-connect these components? What seems odd to me is that I was able to apply different appearance to them (red/white plastic) so that's clearly a property of the component that doesn't propagate to all the other components?
After a little testing, it's clear that I used "paste" instead of "paste new". This happened because I used the standard copy/paste keystrokes instead of the menu (if I used the menu I may still have chosen the wrong type of paste).
So how do if fix this?
Is there anyway to quickly make the copies "paste new" instead of just "paste"?
This is the kind of thing that dives me nuts about F360 - I get that it's all part of the parametric awesomeness it seems these gotchas come up after I've got a couple of hours into a design more often than I'd like.