This is one I'd get the shop to look at, mainly because you need the eyes of an experienced wrench and I don't think advice over the internet is the right thing because we can't see exactly what's happening. A lot of modern chains use single use pins - ie, you can pop them out but have to replace them with a new pin (hence the popularity of quick links - this wasn't an issue with old chains). You can ride it but don't go standing and hammering on the pedals until you've had it looked at.

That 'pop' you felt was the pin pushing out of the plate. When you push the pin back in (as you tried to do), you need to have that plate sitting hard against one of the supports in the chain tool.

Don't sweat on it mate, just get it looked at before being too vigorous on the pedals.

I'm never shy about approaching a mechanic, admitting my stupidity and asking for sage advice, it's an approach that has opened a lot of doors and led to me learning stuff I probably wouldn't have otherwise. Shops that don't support this approach have discovered they don't sell me stuff either.

As for the initial problem. It's not unknown for new chains to have a tight spot and you were right in thinking that it's just the whole plate and pin arrangement being a bit tight. The usual correction is just to grasp the chain in your hands and work it from side to side - that's usually enough to free it. If you need to use the chain tool, just a little pressure then check, then some more, etc.

Richard