Do you know how much the replacing part “the inverter” cost?

You should have mentioned the laptop model. There are tons of different inverters out there.

The service report said the inverter board had been faulty and was replaced.
My laptop had different sypmtoms though, the screen was actually a dull grey/white so it looks like the backlight was working ok but no image appeared.

Your description doesn’t sound like a problem with the inverter board at all. If you can see the screen, even though there is no image on it, the inverter is working properly.
It’s possible that your technician ordered a new inverter board “as a precaution” (that’s how you get parts for test ) but fixed your laptop by reseating the video cable connection.

Screen goes DIM but you can still see everything. Move the screen back and forth rapidly brings it back but now even that doesn’t work.

If you can fix the problem by moving the screen, most likely it’s either bad connection or faulty video cable.
1. Check connections between the video cable and LCD screen, inverter board. Reseat them just in case.
2. If you have the lid close button, make sure it moves freely. Try tapping on the button and see if it light up the screen.
3. Try replacing the video cable, it could be bad. It’s not a common failure, but possible.

When I press the lid close button the laptop goes into sleep mode, so I don’t think the button is “sticking”. Right?

Yep, sounds like the lid close switch button works properly.

I do remember that when I first started having the LCD display problem, pressing on the lid close button would cause the screen to light up and function normally for up to a couple of hours.

Usually, this indicates a problem with the inverter board or backlight lamp.

It’s hard to tell what is causing the problem. I guess it’s possible that your new inverter board or backlight is not good. I’ve seen is happening before.
If you know that inverter and backlight are good 100% then you have a problem with the motherboard. Apparently, it’s not providing power to the inverter/backlight (just a guess).

Sorry, I cannot give you a better answer without testing the laptop with known good parts.

I have a Dell XPS out of warranty.
The screen sometimes changes the color black for red. It is not changed 100% but 80%.
I already tried a connection with another screen and it worked well.
Do I have to replace the screen or is it a problem with this inverter?

This is not related to the inverter. The screen is still bright (even though shows wrong colors), so the inverter and backligth lamp work properly.
This could be:
1. Bad connection between the video cable and motherboard or LCD screen. Try reconnecting the video cable on both ends.
2. Bad video cable. Maybe there is a shorted wire inside the video harness. Try replacing the video cable.
3. LCD screen failure. Replace the LCD screen.

I assume that parts you installed are good.

Take a look at the lid close button if you have it on your laptop. It’s a small button located somewhere close to one of the hinges. Make sure the lid close button moves freely. Sometimes, the button might get stuck inside the case the laptop “thinks” that the display is closed even though it’s opened. Try tapping on the lid close button.

November 5th:
When I press the lid close button the laptop goes into sleep mode, so I don’t think the button is “sticking”. Right? I do remember that when I first started having the LCD display problem, pressing on the lid close button would cause the screen to light up and function normally for up to a couple of hours. However, that only worked for a day or so, and now pressing the lid close button has no affect on the LCD lighting.