The single easiest, most effective thing you could do is shoot raw. (I assume you didn't because you describe the linked JPEG as "the original.") Raw gives you the dynamic range you need to bring the sky brightness down out of the white-clipping range.
Case in point:
Like your cat picture example, this was shot straight into the sun with the sun low on the horizon. It was shot with a Panasonic DMC-GX1, which probably doesn't have as much inherent dynamic range as your Canon 100D. Yet, with a fair bit of exposure pushing and pulling in Lightroom, voilà:
As you can see, I've still lost the sun's disc in the flare. To get the extreme effect as in the cat picture without any kind of exposure blending (manual or HDR), you need to start with either a lot more dynamic range in the exposure or an even more under-exposed shot.
Given the relative camera specs (Canon 100D vs Nikon D7000) I don't think the cat photographer was working with any more dynamic range than is available to you. You just have to tell the camera to save it all by shooting raw, then make use of it with a raw processor. Lightroom is just one of several suitable programs.
Exposure blending methods take a lot more work and are conditional besides. They don't work well when there is motion in the scene, they don't work well without a tripod when there are nearby items in the scene due to parallax, etc.
Your mountain pic is nowhere near as difficult a scene as either my Goblin Valley pic or the cat pic. It may well be that only a polarizer is required.
Here's what a polarizer can do:
That picture was taken by a Canon camera, straight out of the camera with no exposure adjustment. You can tell it was taken with a polarizer by the sky gradient. That's a characteristic effect of a polarizer, especially with a wide-angle lens, since the effect varies with the angle between the sun and the lens axis. A polarizer's effect is strongest at 90˚ to the sun — straight left or straight right — decreasing to essentially zero when the sun is in front of the camera or behind it.
The horizontal angle of view on the lens used for this picture was about 80˚, which is why the effect varies so much from left to right. Since the effect is clearly strongest at the right edge of the frame, that means the sun must be no more than 10˚ out of the left edge of the frame.
I always shoot raw. You never know when you'll need the extra dynamic range to play with.
I also always carry a polarizer, even in my daily-carry bag which holds only 10 lbs of stuff including a tablet and small laptop.