Generally, the goal for using non-antimicrobial soap is simply to pick up and remove organisms and dirt from our skin. This is achieved first by the soap chemically bonding to certain molecules (i.e. oil or lipids). Second, friction from rubbing mechanically and rinsing with water pick up and removes the soap and chemically bonded particles. Thus, the more soap and the longer the hands are rubbed together, the less oil and microbes left on your hands.
For a more scientific answer:
Soap is a detergent, meaning one side of the molecule is hydrophilic (likes water), and the other is hydrophobic (likes oil). Cellular membranes are also made out of a dual sided hydrophobic and hydrophilic molecule called a lipid. The soap solvates the hydrophobic parts of the lipid cell membrane, and you're left with a mixture of lipids & oils suspended in the soap. These are washed away when you rinse with water (imagine how cooking oil floats on top of water).
For another (longer) response to this question see here.
Comments on website credibility:
I'll also comment that at any point that you do an online search, you REALLY need to pay attention to the source of the information you're looking at. The internet is crawling with misinformation and inaccuracies -- often represented as fact or even presented by a so-called 'expert'! Typically, .edu websites (from academic institutions) are MUCH MUCH more trustworthy than .com, .org, etc. because they typically originate from some kind of peer-review or life-long study of a topic.
Be cautious, however, that you don't end up on a .edu page that contains a student's (or other non expert's) thoughts/opinions/or (possibly wrong) response to a question. Further, unfortunately it's not always this simple, as .org or .com info isn't necessarily wrong.
If you want to increase your odds of finding the most accurate information, go straight to the peer-reviewed literature (start a search here). If this is too daunting a task or if you can't find what you're looking for, then I'd suggest focusing on .edu websites for info. You can use the "site:" operator in a Google search to specify a domain -- specifically, if you’re looking for high-quality references, you could use site:.edu to only pull up results from .edu domains.
That being said, I think it's clear that your first link is not nearly as trustworthy as your second, and in this case can be dismissed as internet misinformation.
Hope this helps!