[Enlightened1]

Of interest:

PHI includes health records, health histories, lab test results, and medical bills. Essentially, all health information is considered PHI when it includes individual identifiers. Demographic information is also considered PHI under HIPAA Rules, as are many common identifiers such as patient names

https://www.hipaajournal.com/considered-phi-hipaa/

[Bruce Campbells Chin]

Here's the part you're missing - HIPAA limits what health care providers may disclose, and employers generally don't qualify as health care providers. This is the first link you provided:

https://www.hhs.gov/hipaa/for-individuals/guidance-materials-for-consumers/index.html

And this is quoted directly from your link:

Who Is Not Required to Follow These Laws

Many organizations that have health information about you do not have to follow these laws. Examples of organizations that do not have to follow the Privacy and Security Rules include:

Life insurers

Employers....

In short, HIPAA prevents health care providers from disclosing information without your permission. It does not prevent employers from asking about your medical status, or even from requiring you to provide documentation to support it. It just prevents health care providers from giving that information directly to employers without your consent.