ATMEGA328P-PU and Arduino Nano as Programmer

Again, ATmega328P is important. The 328 differs from 328P or 328PB... Signature have to fit, 328P - 1E 95 0F (this is default Arduino chip), 328 - 1E 95 14 (from MCU's datasheet).

avrdude: Expected signature for ATMEGA328P is 1E 95 0F
Double check chip, or use -F to override this check.

Not a lot. Go to 'Preferences' and switch on detailed log. Post it. We need to know used command, the values in the response etc.
Because we know what expected signature is (according exact part name) but what was received if anything.
BTW, do not use -F, it is not good advice.

Capacitor probably does not help if you have the newest version.

Post the picture of your connection. It might help. Double check your wiring. There are just 6 wires but you would be surprised how often this is the root cause.

Minimal circuit for ATmega must be. With the breadboard it is important but if you are using UNO board it is fulfilled. Oscillator, reset circuit, power - all power pins connected and decoupling. Decoupling is missing in tutorial but many times was proven that it can be important. So, add 0.1uF to close to each power pin, connected between power pin and GND of course.

Finally, it might look silly, but anyway. Do you have the ISP sketch loaded?