Website Insecure Warning: Any way to disable it?

This is a legitimate concern. Google has actually been on this forefront of requiring websites (and web services) to have better security.

Also, there is a setting in:

> Settings > Safari > Privacy & Security > Fraudulent Website Warnings > on/off

I wouldn't advise turning it off, or visiting websites with those kind of issues, regardless.

URL is Uniform Resource Locator. http://jazzfoundation.org/ is an example of one. You will notice that new versions of Safari automatically changes it to https://jazzfoundation.org to load the secure version of that website.

Note the S in https. It means "secure". All websites that have the slightest possibility of transferring information that may be considered sensitive or that are vulnerable to malicious interference should be using transport layer security, at a minimum. If the website's security settings are incorrect, you will encounter that warning.

That particular website loads fine on the Mac I happen to be using, so I'd check your system date and time to be sure they're accurate for your geographical location. If they are incorrect the website's security settings will not correspond to what they are expected to be. That's just a start. Next, examine your Mac's Network preferences for any unexpected proxy or DNS settings. Additional information might be revealed by using EtreCheck: Using EtreCheck - Apple Community

I deselected the option to warn me when visiting a fraudulent site,

Totally different subject. Fraudulent websites are those specifically blacklisted by Google's Safe Browsing service, which Safari uses. Leaving it on is harmless and generally beneficial.

If this describes what you're encountering:

Avoid fraud by using encrypted websites in Safari on Mac ...

... then the warning is legitimate.

The correct action is described in that Support document (boldface added):

• Contact the website owner or administrator and ask why their site causes a certificate warning. For example, they may say the site is only accessible within your organization, so it has a self-signed certificate (one not provided by a certificate authority). You have to decide whether to trust the site anyway or not visit it.

Saxman Said:

"There are far too many websites Safari deems insecure, and will show a warning window instead of taking me to the site[...]"

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Scan for Malware:

With Security & Privacy being of high-concern here, scan your Mac for Malware (Malicious Software). Such software configures your Mac to act in a mislead manor.

Use MalwareBytes for Mac: This is software that searches for malware/adware. So, scan with it and then remove what is found. Once removed, uninstall MalwareBytes for Mac. Then restart the Mac.

Downloads:

1. Malwarebytes Anti-Malware for Mac
2. Malwarebytes uninstaller

When that warning pops up so frequently, and is wrong that often, it loses it's credibility, IMHO.

You're absolutely right. Warnings that become a nuisance degrade security. Apple knows it and avoids needlessly throwing them in your way, so when you encounter something like you're describing you should absolutely take it seriously.

I suspect something else is going on. It could be your router or a related Network setting. Recent scams have been known to leverage them but what you need is a consistent example so that you can bypass those unknown factors.

DNS servers, how am I supposed to know if the ones shown are correct, or which ones I do want?

Unless you have reason to provide DNS Server addresses and are certain what they should be, then let your ISP provide them. They determine what's best for their subscribers, not Firefox. Same goes for Proxies. Normally there should be none.

System Preference panes are grouped by category. To view an alphabetically sorted list click the View menu.

So many questions. You should Contact Support for answers. There is reason to believe you misconfigured your system in some manner.

Can you provide an example?

I'm still waiting for you to provide a representative URL. Until you do that all anyone can do is speculate.

Saxman Said:

"The issue is that I get these pop-up windows, requiring me to click through several more, and input my administrator password, on far too many sites, some are sites I go to quite frequently, and they all warn of the risk that it's not really the site[...]"

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Taking Apple's Word for it:

Apple is Apple --- they know what is, and what is not, safe for Macs - particularly Phishing Sites and Scam Sites in this case.

So, I'd stay away form those Web sites.