The requirement for holding a first class medical is in the FAA FSIMS, the updated version of the inspectors handbook.
The section quoted is for ATP applicants engaged in operations under Part 121, 135 or 91 Subpart K. For the life of me I cannot figure out why they have that section in there since there is NO regulatory requirement to have any level of medical certification for any knowledge test, regardless of whether you're engaged in Part 121/135/91K operations or not. When we had first-time ATP applicants at my 121 carrier, all we would suggest they do is go get their written done at a testing center prior to their checkride. My thought is it is old and outdated guidance that has not been removed/changed yet.
The FAA knowledge test pre-requisites can be found at 14 CFR 61.35 and are very basic: An instructor endorsement (if required for the certificate/rating you are testing for and there is NO INSTRUCTOR ENDORSEMENT required for an ATP written per 14 CFRR 61.153) and a valid photo ID.
FAA knowledge tests are generally administered at a computerized testing center. To take the ATP written all I did was register and provide ID when I showed up. No one asked for logbooks or medical certificates or anything and I'v never heard of anything like that happening.
When there is a discrepancy between the regulations and FAA guidance, the regulations always trump. FSIMS is not regulatory and is meant to guide FAA Inspectors in the performance of their jobs.