USB-Port1 --> USB-to-Serial-Converter --> RX, TX --> ESP-01 TX, RX
USB-Port2 --> 5V, GND --> LD33V --> VCC/CH_PD, GND
This setup runs fine. I get serial connection at 9600 BAUD and send AT commands.
Now my "problem". If I replace USB-Port2 by an usb powerbank or a 5V USB wall adapter I can see the ESP booting on serial terminal but it doesn't respond to my AT-commands. When I switch back to USB-Port2 all is fine again.
I tested one powerbank (output 2A) and different USB-wall adapters (500ma, 1A, 1.5A) all of them produces the same "error".