It looks like the first two types of hacking you mention in that webpage take place outside of my computer \[...] So in those cases, I guess I don't have to worry about someone accessing the data on my computer?
Correct, but even in the case of the trojan, that isn't the primary concern. The data they seek to steal from you would be entered on a web site by you, if you didn't notice that it was bogus. For example, a visit to www.paypal.com might be redirected to a PayPal-lookalike scam site designed just to get you to provide your username and password. That danger is the same regardless of the source of the problem.
As to the issue only appearing on one machine, it depends heavily on the circumstances. DNS cache poisoning is a purely transitory problem... a hacker gets incorrect entries added to the DNS server through trickery, but those entries are eventually replaced as the cache is updated. It's possible that by the time you tried the other machine, the problem had resolved itself.
In addition, these spoofed entries only map certain specific domain names to malicious IP addresses. If you did not visit a site with a poisoned DNS entry on the second machine, you would not see any difference from normal.
It could also be a trojan... that's easy to rule out with something like [ClamXav|http://www.clamxav.com>. Any trojan should be viewed as a serious security breach and responded to appropriately, but in the case of RSPlug, I don't believe it does anything other than DNS redirects.
When I get home, I'll try one of those scanning tools. But, I'm not sure how to tell if they're working or if the problem is still there given that it stopped happening once I clicked on their "continue to internet" link.
If you find a trojan, things are easy... just remove it and you're good. (With RSPlug, simple deletion isn't adequate... there is a free removal tool [here|http://www.dnschanger.com>.)
If there's no trojan involved, things become a little more complicated. You should probably contact your internet service provider to let them know what happened. They may be aware of the issue and can tell you what they have done about it. You would also be wise to change the password in your router, and verify the DNS server settings in the router. Changing the DNS server in your laptop or your wireless router might not be a bad idea as well. Try either the [Google DNS servers|http://code.google.com/speed/public-dns> or [OpenDNS|https://www.opendns.com>.