The laptop would probably handle the cameras if they were low resolution and doing no motion detection, i.e they were recording 24/7 (which is not a problem for a laptop - they can run 24/7 as well as any desktop). Motion detection done by BI will be what uses the most CPU. If all you want motion detection for is alerts, many cameras can do this internally themselves and don't need BI (or any NVR software) to send you an email. Recording 24/7 will use the least CPU and memory resources, so unless you want recordings that go back months, recording 24/7 will be the easiest and least computer intensive.
You really should just download the FREE trial of BI at the link given in my previous post and try it out - then you'll KNOW now instead of guessing.
For video clip storage an inexpensive (around $100) external USB3 2 to 4 TB hard drive will give you plenty of time for practical use at home - highly recommended.
I have no practical experience with the D-Link camera line, however, why do you want to get a camera with the same resolution as your Logitechs? You might as well keep them if you aren't interested in 1080p + video resolution. I would think you would be looking for at least 2-3MP cameras (anything that can do at least 1080p at 30 fps). There are several cameras like the D-link you mentioned that can do better.
If you sell your 6 Logitechs you'll have about $2,000+ for your budget - this will buy you a VERY nice system - even professionally installed, and with that budget I would seriously consider getting a full blown 8 or 16 channel NVR, it would be MUCH easier to set up and use than a PC with BI (or any NVR software for that matter). Since ease of use is one of your primary goals I think a real NVR would be a better fit.
There are several professional security camera installers on this board and I'm sure any of them would be able to supply you with a "plug and play" sytem on your budget. I would also weigh the advise given by @
fenderman and the handlful of other professional here on the board on camera choice - he is one of the more experienced members on this board with cameras and NVR's and has practical experience, which is very helpful - this is why I've been asking him about camera choices (in several threads on the board).