Juraj:
what does "AT_UART_DEF?" return.
it is because you flashed a 2 MB firmware to a 512kB flash?
This is what i get:
AT+UART_DEF=9600,8,1,0,0
OK
I think that for my ESP-01 model, it has 512+512 = 1 MB flash? During flashing, this is the detected info:
flash vendor:
E0h : N/A
flash devID:
4014h
QUAD;8Mbit
crystal:
26 Mhz
I tried again with flash size of 4Mbit, but it goes in a looping sequence for baud 74880. I did some trial and error and went back to the earliest AT firmware available: 1.5.1 and i used these settings which finally worked for my ESP-01 model.

I used ESP8266 AT Bin V1.5.1: https://www.espressif.com/sites/default/files/ap/esp8266_at_bin_v1.5.1.zip and these are my settings from the flash downloader tool:
0x00000 boot_v1.7.bin
0x01000 at/512+512/user1.1024.new.2.bin
0xFC000 esp_init_data_default.bin
0xFE0000 blank.bin
First, i entered AT+UART_DEF=9600,8,1,0,0 at baud rate of 115200 and got OK response. Then, at baud rate of 9600:
AT+GMR
AT version:1.5.0.0(Oct 24 2017 12:03:18)
SDK version:2.1.0(ace2d95)
compile time:Oct 24 2017 15:48:02
Bin version(Wroom 02):1.5.1
OK
AT+RST
OK
c_⸮RS⸮fJ⸮:f⸮[:fJ⸮JO⸮⸮s⸮S⸮⸮X⸮⸮⸮⸮⸮⸮
ready
AT+GMR
AT version:1.5.0.0(Oct 24 2017 12:03:18)
SDK version:2.1.0(ace2d95)
compile time:Oct 24 2017 15:48:02
Bin version(Wroom 02):1.5.1
OK
Software reset seems to store the baud setting and so does a hardware reset. I think that i've finally found a winning combination!
Edit: I went through the official AT instruction set document: https://www.espressif.com/sites/default/files/documentation/4a-esp8266_at_instruction_set_en.pdf
and i found this section which is relevant to my particular ESP-01 model:

I selected the firmware files in the same order as shown above and hopefully flashed it properly.

AT
OK
AT+RST
OK
c_⸮RS⸮fJ

