I did an inadequate job of trying to explain this my first attempt. Let me try again.
The systems in your pictures above are both functioning exactly as they are supposed to and do in reality.
You have to understand there are four basic types of instrument approaches. VOR, LOC, ILS, and GPS. I will try to explain how each work.
1. VOR: This is an omnidirection transmitter that is capable of providing navigation signals on all 360 degrees of possible approach to and from the station. This is why it can be used for both enroute navigation as well as for an instrument approach. For an instrument approach, you normally see the VOR navaid closest to the runway chosen to contruct an approach, but sometimes you have to use the one a bit further away but oriented closer to runway orientation (meaning you can use it to head to the approach end of the runway or a radial that is reasonably close to runway orientation).
It is a non-precision approach that only can provide only course guidance, which in the case of a VOR is done by dialing in the closest radial setting to match runway orientation. However, among the many factors that make VOR approaches the least accurate of them all, is that fact often the runway is located and oriented in a way that cannot precisely match even the closest available radial. So, you sometimes see a VOR approach where the desired radial is up to four degrees off the runway orientation. So, you come in at an angle and need to reorient your aircraft when you break out of the clouds and see the runway. If you fail to set the proper radial as depicted on the approach plate (sheet of paper) for the given approach, you will be trying to center up on a totally inaccurate courseline. Remember, on a VOR approach, the course you center up is for the radial you dial in on your CDI or HSI.
2. LOC: For a localizer approach you are not using a VOR station. Instead, you are using a station located in close proximity to the given runway. This station is not like a VOR since it does not provide omnidirection beacon signals, the station is pre-set with just one course setting. In other words it transmits its course bearing on just one line of radial, and therefore you can orient yourself either toward that radial or away from it. If you use the reverse radial, then you are flying what is called a "back course localizer."
This is why you can spin the CDI's compass rose (the circular band of headings) or the HSI's course line (the yellow line) to any heading you care and it doesn't make any difference to the accuracy of the course deviation indicator. As in your example pictures above, you have the HSI course line clearly oriented about 30 degrees off the runway orientation, but since you are actually on a straight line orientation from the runway heading, your course deviation bar shows you pegged on extended runway centerline.
3. ILS: Instrument Landing System: This is similar to a localizer (LOC) approach except it features an additional glideslope indication. So, not only does it reference a fixed course radial, it also is tailored with a fixed glideslope beacon, essentially a signal oriented from the normal runway touchdown location on about a three degree azimuth above level ground.
Again, the course line on the HSI and the compass rose on the CDI can be set to any value possible and it doesn't make any difference because the associated fixed degree radial of the ILS is set for the runway orientation. The glideslope is also fixed.
For all three of these approaches, the dots related to a known deviation in course and glideslope (in the case of the ILS only). So, when you are a dot off to the right, it doesn't mean you are a certain distance off ideal alignment to the right, it means you are a certain number of degrees in error.
4. GPS: Global Positioning System: In this case, your aircraft's GPS location is mathematically computed relative to the ideal location for the given GPS approach. Further, with Wide Area Augmentation System (WAAS) your GPS approach uses an additional ground station to suppliment and rate the accuracy of your raw GPS satellite position interrogation. With WAAS, you now have the ability to rate the vertical deviation accurately enough so that the GPS approach can add a glideslope input to your instrument approach. Of course, this can only be safely done if the WAAS and GPS signals are sufficiently strong and clean enough.
Here is a link to discuss more details:
http://en.wikipedia.org/wiki/Wide_Area_Augmentation_System
So, when these required qualities are met, your GPS receiver indicates Localizer Performance with Vertical Guidance (LPV), which is the highest quality of signal strength allows you to accurately fly a localizer and glideslope referenced instrument approach. Just like with the LOC and ILS approaches, it doesn't really matter what course you dial into your CDI or HSI if you are using an advanced GPS Steering computer. I have an STEC System 30 autopilot on my Skyhawk that is this way. So, for a GPS approach to runway 31 (310) I could if I wanted to dial in 270 into my CDI and the autopilot will still fly the course just as accurately as she pleases.
Ironically, my 310R has an older Cessna 400 autopilot and it is tied directly into my Bendix King HSI and so I do have to dial in the desired courseline for the autopilot to work properly when flying GPS inputs, either for an approach or for enroute navigation.
However, if I manually fly the approach, which I do as habit, it doesn't matter what course I dial in to my HSI. The course deviation bar shows how far left or right or course I am and that course reference will be extended runway centerline.
OK, so to review ...
VOR approach, you have to dial in the proper course into your CDI or HSI or else your approach will be off course of the runway, and in the real world is life threatening.
For a LOC, ILS, or GPS approach it doesn't matter what course you dial into your CDI or HSI, the course deviation bar will remain fully accurate. You'll simply have to look at a course line that isn't pointed up and down on your HSI. And on a CDI, you won't notice anything abnormal at all.
Hope that helps to explain it.
Cheers,
Ken