klabautermann Logically, the missed calls list only contains the calls when the phone was switched on.
Yes, this is expected. The dialer is not working to register the missed calls.
Some mobile operators send text messages (SMS) with info about missed calls. I know what you mean, I previously used such service.
And it's the same as what @NetRunner88 advised - this is an operator/provider feature, not a device one. There is no other way when the phone is off.
But what I noticed is, that if I go to airplane mode (or turn off the phone) and then call my number from another - I hear rings like usual and it does not tell me the number is not available or goes to voicemail. It's misleading for the caller who may think I'm not picking up but getting the missed call registered. I've tried multiple times in a 5 min. interval. I'm using 4G-only mode (not sure if that matters). Maybe when the phone goes to airplane mode or turns off it does not tell the network, so the service provider needs more time when my phone hasn't reported to consider it is off.
So my assumption - maybe your old phone unregisters itself from the network during the turning-off process.