I looked at the US Embassy in Germany website. You are correct. It says 6 months. It is 100% wrong and has old information.
I have given you the reference for the German Embassy in the US website. Now, I'll give you a URL for COVID-19 entry rules on Germany's Federal Foreign Ministry's website: https://www.auswaertiges-amt.de/en/coronavirus/2317268
Click on Box #3 and scroll down to...
Proof of recovery: A positive PCR test result carried out at least 28
days but no more than 90 days previously.
How about the German Federal Ministry of Health (https://www.bundesgesundheitsministerium.de/en/coronavirus/infos-reisende/faq-tests-einreisende.html)? Search for the word "recovery". At occurrence #7 of 16 on the webpage you will find...
What constitutes proof of recovery? Proof of immunity acquired through
previous infection with the SARS-CoV-2 coronavirus issued in written
or digital form in German, English, French, Italian or Spanish.
This proof must comply with the following criteria published in
section 2 number 8 of the Ordinance on Coronavirus Entry Regulations:
-direct pathogen detection of the prior infection,
-The test to prove the prior infection was performed no less than 28 days
and no more than 90 days earlier.
So, there is 100% consistency among German government websites about what constitutes recovery. That should remove all doubt about what the rules for recovery are.