Facharzt would be board certification / board certified, aand facharztähnlich means that the physician has gained experience equivalent to board certification level.

The list stipulates certain requirements, and specialists who don't meet these requirements may not be added to the list, is the upshot of the meaning here.

Simply "specialists" (works for both UK and US)

(doctors who are not specialists or of similar professional rank)

Since you only seem to need the second part of "ohne Facharzt/ facharztähnliche Qualifikation" the following link shall provide a number of solutions :

https://service.berlin.de/dienstleistung/331212/en/
e.g.:
/recognized equivalence of professional qualification
/recognized equivalent level of education and training

"The professional qualification is equivalent if there are no significant differences between your foreign professional qualification and the German professional qualification.
If your specialist qualification is recognised, you can work in the profession in Germany. You will receive a notification."

(Facharzt can be translated as medical specialist or simply specialist.)
https://www.anerkennung-in-deutschland.de/html/en/2706.php
The latter link also mentions the need to have one´s qualification recognized in order to practise in Germany (most likely the same procedure in other countries too).
:"If you have attained additional medical job specialisation and a qualification as medical specialist, you can apply for recognition of this specialised professional title in Germany... Please note: You need approbation as a doctor of medicine before applying for the recognition of your specialised title."

In short: simple doctors of medicine and doctors, who cannot produce a recognized proof of equivalence of professional qualification as a specialist, are ruled out from certain jobs/duties.