DVI adapter for a 2x15 pin LCD plug
There's no guarantee that the interface from a laptop's graphics chip to the internal display panel is the same interface as the one to the external display. The part from the manual:


  The 15-pin D-sub monitor port supports a standard VGA-compatible device such as a monitor or projector to allow viewing on a larger external display.


Is almost certainly not talking about this internal connector (which has twice as many pins!), but instead talking about a 15-pin D-sub on the exterior of the laptop. You know how there are a lot of graphics cards with one DVI output and one VGA output , both of which can drive a display? Laptops would use the DVI interface for the internal and the VGA interface for the external. There is no implication of compatibility or interchangeability between the two.

Internal monitor interfaces were DVI-based for a little bit several years ago, but are now mostly LVDS-based interfaces. There is not enough information to tell which from your question, but in general, odds are that that connector is LVDS, does not have a standard pinout, and does not have VGA or DVI signals.