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Regarding your question, I will try to explain the paragraph you linked above from what I learned during my research.
As stated in §18a of the Asylum Act, people who claim asylum in the transit area of an airport who either come from so-called "safe countries of origin" or do not have valid identification face a special asylum procedure at the airport. The decision on their case has to be made before they are allowed to enter Germany. That is, if the applicants can be accommodated at the airport during the procedure.
Technically, this paragraph doesn't speak of people with valid passports from countries which are not on the list of "safe countries of origin", who claim asylum at the airport.
In those cases, it is up to the officials from the Federal Office for Migration and Refugees (BAMF) if they decide to let you enter Germany right away and bring you to a reception center, where you can go through your asylum procedure, or if your asylum claim will be handled at the airport directly.
This is also explained by one of the sources I found:
"The airport procedure usually applies to applicants who do not have valid documents upon arrival at the airport, but it may also apply to applicants who ask for asylum at the border authorities in the transit area and to those who come from a “safe country of origin”. In practice, however, it is not applied to unaccompanied children."
If you scroll down on the same website, four different potential outcomes are described:
The Federal Office decides within 2 calendar days that the application is “manifestly unfounded”: Entry to the territory is denied. A copy of the decision is sent to the competent administrative court.2 The applicant may ask the court for an interim measure against deportation within three calendar days;
In theory, the Federal Office can decide within the 2 calendar days that the application is successful or it can reject the application as “unfounded” (unqualified rejection). In these cases, entry to the territory and, if necessary, access to the legal remedies of the regular procedure would have to be granted. However, this option seems to be irrelevant in practice since the Federal Office always grants entry to the territory for the asylum procedure to be carried out in a regular procedure if an application is not rejected as manifestly unfounded;
The Federal Office declares within the first 2 calendar days following the application that it will not be able to decide upon the application at short notice. Entry to the territory and access to the regular procedure are granted; or
The Federal Office has not taken a decision within 2 calendar days following the application: Entry to the territory (and to the regular procedure) is granted.
According to this source, most outcomes lead to the entry of asylum applicants in order to further go through the asylum procedure and seek legal support. In 2017, number three has been the most common outcome according to the website.
There is further information of the airport procedure in Germany by the BAMF here: http://www.bamf.de/EN/Fluechtlingsschutz/Sonderverfahren/FlughafenVerfahren/flughafenverfahren-node.html
I hope this explanation helps. Please reach out to us again if you have any more questions. To all the other members on this platform: feel free to share your experiences, too!