This is from panel-mipi-dbi-spi.txt
command 0xE0 0x00 0x03 0x09 0x08 0x16 0x0A 0x3F 0x78 0x4C 0x09 0x0A 0x08 0x16 0x1A 0x0F
command 0xE1 0x00 0x16 0x19 0x03 0x0F 0x05 0x32 0x45 0x46 0x04 0x0E 0x0D 0x35 0x37 0x0F
This is from LCDWIKI_SPI.cpp for ili9488:
0xE0, 15, 0x00, 0x07, 0x10, 0x09, 0x17, 0x0B, 0x41, 0x89, 0x4B, 0x0A, 0x0C, 0x0E, 0x18, 0x1B, 0x0F,
0xE1, 15, 0x00, 0x17, 0x1A, 0x04, 0x0E, 0x06, 0x2F, 0x45, 0x43, 0x02, 0x0A, 0x09, 0x32, 0x36, 0x0F,
They are other commands that differs slightly.
I've found other sources for the init sequence which seems all to be close to panel-mipi-dbi-spi.txt :
* Juj fbcp lib: https://github.com/juj/fbcp-ili9341/blob/master/ili9488.cpp#L28
* Bodmer eTFT_SPI lib for microcontrollers : https://github.com/Bodmer/TFT_eSPI/blob/master/TFT_Drivers/ILI9488_Init.h#L11
* Jarret Bucket Adafruit GLX lib : https://github.com/jaretburkett/ILI9488/blob/master/ILI9488.cpp#L378
* this one is for STM32 : https://github.com/RobertoBenjami/stm32_graphics_display_drivers/blob/master/Drivers/lcd/ili9488/ili9488.c#L219
In the same manner, we can found init sequences for other chips in these libraries, eg ili9486.
There's already a driver in the kernel source : https://web.git.kernel.org/pub/scm/linux/kernel/git/stable/linux.git/tree/drivers/gpu/drm/tiny/ili9486.c?h=linux-6.12.y but it is specific to Waveshare 35a or PiScreen displays (these with 4 kind of shift register ICs onboard).
So for now, while waiting for a new ili9488 display arrives home, I'm digging further around the right init sequence for ili9486.