It seems as though it is "missing" the red or blue colour between the inputs and the output of the projector. The green is OK because the output display is tinged green.
(All colours are made up of a combination of red, blue and green colours of varying values, e.g white occurs when you have all 3 signals on and all having a 100% value).
It can be a hardware problem as all the signal inputs pass through the same input signal processor stage.
The menu display and the blue no signal screen, being internal, are probably injected further into the signal path towards the output and therefore do not go through the input signal processor stage to get to the output. In a way this is good news as it means that the colour filters and motor etc. in the output stage are OK.
Try connecting a laptop so that you can view a single colour, i.e. have only a red screen on the laptop display and see if it shows through the projector as red. Do the same with a blue screen. They may look slightly different as you probably cannot get a pure colour, but initially it should suffice to indicate which one is working and which one is not. Use this link on your laptop when connected to the projector. Click on the colour to make it go full screen. (Click in the colour 'box outline' to go back to the menu).
Here is a link to the service manual for your projector.
To download the file, when the sentence below the Document preview box changes from This file is downloadable free of charge: ..processing.. to This file is downloadable free of charge: Get Manual click on the Get Manual as this is the download link. Be patient it takes a few minutes to find the file.
Hopefully this is of some help.