I am looking for monitor, which can not emit light - something like old Nokia's display, only bigger. I need it to have a backlight, which could be toggled - just like mentioned old phone display. How are these monitors called? I would need 4-10 inch size monitor.
Depending on the requirements of your particular application, you may find that an e-ink monitor will suit your purpose. E-ink displays are slow to update, which means that video and animated images will blur/smear. E-ink devices such as the Amazon Kindle require external illumination, although some now include internal (reflective) illumination.
Amazon has a number of small (13") monitors that use e-ink technology. The selected monitor has a touch screen and also built-in illumination (with on/off switch) but there are others listed without those features. I used "e-ink computer monitor" as the search terms to locate these devices.
Additionally, one can find Arduino and Raspberry Pi accessory devices which use e-ink technology. Sized much smaller than 13", they are also limited in providing moving images.
I use an older Kindle Reader without internal illumination options. It is larger than your desired limit, but newer models are closer to your requirements. There is no feature to allow external input, but it is referenced as an example that the size you seek is likely to be found.