Paris (my city): you already noticed than bus and metros have different prices.
You can transfer from metro to metro or metro to trains, it doesn't charge another trip if you have to pass barriers, unless there's a system failure...
Apart from the monthly pass, us locals have a kinda new offer (2 years ago) that allows all transfers, should have been like that for ages!
Yes, two apps, but you can buy a 2€ plastic card instead.
Knowing what and how to recharge this card (or the app) would need a whole post.
Munich (recent holiday)
close to what you described for Warsaw: buy a weekly pass from a machine, it comes already stamped, the ticket won't leave your wallet ever again (no need to show it unless there are controllers). There are zones and you can buy extension tickets. And there is an app too.