This is controlled by Android. Sounds like you've set one browser as the "default" for http, and the other one for https.
Please open the device's system settings, then Applications, then open Dolphin and the built-in browser in turn, and press "Clear defaults" for both (may need to swipe or use the menu to see the HTC browser in the "System" category).
Now the next time you tap a link, Android will again ask you to choose which one to use. You'll be able to set one as the default, and be sure to use the same one.