Should be the other way around. Money taken out for known expenses should be on par with money for bills.

I assume he meant known, yet undetermined, expenses, which is sensible. For instance, you know that your house and car are going to need repairs eventually, so it makes sense to have some money set aside for those rather than dipping into your "emergency fund" -- otherwise what happens if a real emergency occurs before you finish paying it off?

Once you have enough money in your savings, though, I think you can stop thinking in these terms.

Yeah, it's stuff you know you'll need money for that won't be an emergency. Just like saving for anything else (like a new laptop).

I took 'on top of an emergency fund' to mean that the money being set aside for those things was not part of the emergency fund.