More of a photography question than an Android question, I think. Remember that HDR is High Dynamic Range, essentially your camera takes 3 (or more) pictures at different exposures: One normal, one bight, one dark (EV -1,0,+1). It then combines all of these to create an image that represents what 'what your eyes see' by bringing out the dark parts of the photo with the light copy and dimming the bright parts with the dark copy (a simplified explanation).
With the images you have shown me, it seems as if you are shooting directly into a light source, and while this works if you want to shoot a silhouette, it's not the best option for HDR style photography unless you have control on how to edit the Highlights/Shadow -essentially Photoshop type post-processing. This is not very conducive to HDR imaging, and thusly will produce 'bad' (not desired) images. I would suggest taking a picture where the light source is behind the phone (not aimed at the lens) as a HDR comparison.
To explain for the difference between Stock Camera HDR and Best HDR Pro, my thoughts go to the HDR algorithm. I assume that both apps have a different way of processing HDR images, and from the looks of it, the Stock Camera is using an algorithm that avoids the use of noise while Best HDR Pro is pushing either ISO or brightness up to give a more HDR like look.
Like I stated before, take images with light coming in from different directions and try to take images with quite a bit of detail (flowers, landscapes, ect.) to get a good comparison. That should help you decide on the performance of the apps themselves.