Soldering a VGA cable - number of wires doesn't match
You actually have 14 wires if you count the ground return for each colour, it may use one of those on pin 5 (GND) as in your diagram to complete the set of 15. The other pins are often used for communications. For example from VGA connector on Wikipedia some of the ones your diagram showns as not connected are:


Pin 4 - ID2/RES - formerly Monitor ID bit 2, reserved since E-DDC
Pin 9 - KEY/PWR - formerly key, now +5V DC
Pin 11 - ID0/RES - formerly Monitor ID bit 0, reserved since E-DDC
Pin 12 - ID1/SDA - formerly Monitor ID bit 1, I²C data since DDC2
Pin 15 - ID3/SCL - formerly Monitor ID bit 3, I²C clock since DDC2


The reference to DDC refers to the Display Data Channel that the host can use to query the monitor for its capabilities such as supported resolutions and refresh frequencies. So while I believe in many cases they could be left disconnected with the loss of that functionality they won't all be additional ground signals as you mentioned in a comment.

If you're feeling experimental (and there are no guarantees this won't cause damage) because I'm not sure if the colours are 100% standard I'd try the following:


Connect the RGB center wires to pins 1/2/3 with the shields going to pins 6/7/8 respectively.
Short pin 8 to pin 5 to get the additional ground.
Connect the other wires as per your diagram leaving unknown ones disconnected


Now see if you can get an image on the projector by forcing a resolution you know that it supports because you won't have the DDC channel. If that works OK and it's not a problem leave it at that. Otherwise you could probably identify the remaining pins by measuring between ground and the DDC2 pins 12 / 15 using a scope during initial negotiation to check for the square clock versus data. The 5V to go to pin 9 should be easy enough to spot and is presumably present the whole time.