I know this will sound off-topic, but it might be worth mentioning..
The lighting in your top two photos aren't particularly bad or anything, and don't seem far off the white dog pics. It's certainly not different enough to warrant it being an issue. The ''edge'' that the pictures in the link have is more to do with the rest of it. White dog with complimenting dark background (the brown background doesn't work nearly as well IMO). Also, the white dog is always doing something (salivating over biscuits I should imagine). There's also more negative space with the little white dog, making it seem more ''free''' within the space, whereas maybe your own background doesn't give you that freedom. With hers, there's enough space for it to feel like she could be in a park or something, which helps the liveliness.
If you managed a wider field of view and teased your dog into a snarling frenzy, and used a background colour more suitable for the dog, the lighting would appear much more dynamic. The white dog pics have the benefit of a very reduced colour pallette, which you won't achieve without painting your dog another colour.
I'd personally set up the lights/background (big background), make the dog go a bit crazy with the help of an assistant and some dog treats (she MUST have used an assistant for this?) and snap snap snap away. The more fun the better the photos will be. I think the success with the white dog pics is the variety of actions and ''looks'' that the dog is giving, as well as a variety of portrait/landscape orientations.
In the white dog's last picture, it's a good, close shot that makes the head look unusually large.. and that makes the other pics seem better too. It's the full package rather than just the lighting.
Softbox or umbrella? On a personal level I'd go for an umbrella just because it's lighter and simpler.. looking at the dog's eye reflection in the bottom photo it looks like the white dog was lit with a large rectangular softbox.. but don't let that bother you too much!