Thanks for your very helpful post again! I guess my original issue is fixed.
Now it just left me with a small issue (side-effect):
Yes, I use Pi Hole as local DNS server now also and via Conditional Forwarding it is getting the names from the fritz.box. However, the resolution includes always the fritz.box-suffix.
For example: KodiPi.fritz.box (in my client section within PiHole).
Do I still need to use the Local DNS Records to "correct" the names manually? Because I don't want to access my devies via "KodiPi.fritz.box", I want to access it via "KodiPi" (without the suffix).
Is there any solution for the Conditional Forwarding that it uses exact the same names w/o fritz.box suffix?
Example how it looks like:
Fritz.box:
PiHole:
Thanks!