I'm no expert, and I'm sure most here can give you a better perspective than I can, but from what I've learned and experienced the hard way, for the most part, you don't want to clone your hard drive if your intent is to use it for another system. What I mean is that, you're planning on cloning your laptop hdd, which has its settings and OS installation geared towards working in favor of your specs, hardware setup, BIOS config, etc...So you're better off just installing the OS from scratch and then working with the backup that you have.
I understand the time this would consume, and the reason you would want to maintain everything as is.
A couple of years back, I cloned my hdd of my Sony Vaio desktop and copied it to the hdd of my HP pavilion desktop. After 3 or less reboots, the entire OS went haywire, and I could not boot back into windows for the world. Apparently, and from what I was told, read, it had a lot to do with the BIOS(and the fact that they're obviously different between the two vendors I chose to swap with) so in that sense I was left with no choice but to start anew. Thank god for external hdds and the whole concept of backing up every two seconds.
As for needing an external hdd, I don't think that is necessarily the case. I would agree that it's wiser as you could backup both the image and the raw data just in case anything went wrong with the cloning process. Theoretically though, I don't see why you can't just place your laptop drive in your desktop(assuming it's SATA, if not you'll need an IDE---> SATA Convertor found for like $6) and backup the image as well as burn it to the existing drive. I could be wrong with all of this, and I'm sure christ, smokes, or dweez(forgive me if I forgot anybody else) would be able to give you words of wisdom.
As for the app, I can't remember the one I used and it's driving me nuts. I have used Acronis's suite, albeit not for cloning my hdd, but I must warn you that the UI is a bit annoying if not confusing. It's also not the smartest app out there for this. If I come across the name of the other one I'll let you know. Good luck.