Problem while burning a bootloader to an Atmega328p (not in sync)

Hello,
I have some Atmega328p chips and I want to burn a bootloader on each of them. Although I ordered them with a bootloader installed, when I connect an LED between pin 13 (I tried pin #13 and the pin that corresponds to GPIO13 on an Uno) and GND it's not turned on so I assume there's no bootloader on them.
My problem is that I'm not able to burn a bootloader on the Atmega although I follow the steps on https://www.arduino.cc/en/Tutorial/ArduinoToBreadboard.
Unfortunately I have to use the internal clock because for my application there isn't enough space for an external one (plus I'm not that good at soldering so I try to solder as few things as possible). I wired everthing correctly but the Arduino IDE always shows me the following error:
avrdude: stk500_recv(): programmer is not responding
avrdude: stk500_getsync() attempt 1 of 10: not in sync: resp=0x03
... (attempts couting up) ...
avrdude: stk500_recv(): programmer is not responding
avrdude: stk500_getsync() attempt 10 of 10: not in sync: resp=0x03
As a working Arduino I use a Leonardo, IDE version is 1.6.13 (because on the link above there's no download for 1.8.x). In the ArduinoISP sketch I uncommented #define USE_OLD_STYLE_WIRING because the Leonardo doesn't support SPI on the digital pins.
My wiring:
Leonardo --> Atmega328p
GPIO 10 --> #1 (RESET)
ICSP 1 (MISO) --> #17 (MOSI)
ICSP 2 (VCC) --> #7 (VCC)
ICSP 3 (SCK) --> #19 (SCK)
ICSP 4 (MOSI) --> #18 (MISO)
ICSP 6 (GND) --> #8 (GND)
GND --> GND
5V --> VCC
Does someone know why I get these errors? When I use other sketches like Nick Gammon's board detector or fuse calculator it also won't work (same wiring and also same Leonardo).

MISO goes to MISO
MOSI goes to MOSI

Those abbreviations are Master In Slave Out, Master Out Slave In - so they do not get crossed like serial.

That said, I am surprised that that's the error you're getting for that (i'd expect target not responding or something, while your error indicates a problem talking to the

I changed it and I still get the same error.
Here’s a quick draft of the setup (Although there’s an Arduino Uno in it I use a Leonardo):
EDIT: I also tried the same setup 1. with another Atmega and 2. with the Arduino IDE 1.0.5 (because down below on the page it says “Attention: This procedure works on Arduino 1.0.x software.”). On this version I get the error avrdude: stk500_getsync(): not in sync: resp=0x00.