Building notification who has experience with it

but the architect was paid for that, to prepare the documents in such a way that a building permit is obtained,

So we are talking about a fully qualified independent architect, not a draftsman for the builder? In any case, I associate the term building notification with a simplified procedure / "exemption" (and the assurance to build without deviations according to the specifications of the development plan). How this possibility is supposed to be opened on the one hand, but on the other hand cannot be fixed by an amendment, removing a window at an exception-required location and adding one in a location without exceptions, is, to put it mildly, beyond me. If, moreover, a forestry authority also has to give its input, I see even less a simplified procedure as possible. For what actually: is a wooden house to be built on the edge of a forest, or is a tree-fall boundary to be confirmed?

So we’re talking about a fully qualified independent architect, not a draftsman for the construction company?
I associate the term "building notification" with a simplified procedure / "exemption" (and the assurance to build without deviations according to the specifications of the development plan). How this option on the one hand is supposed to be possible, but on the other hand cannot be remedied by a correction to remove a window at a location requiring an exception and to add it at a location free of exceptions, is, to put it kindly, beyond me. If in addition a forestry authority also has to have its say, I see even less possibility for a simplified procedure. For what purpose actually: is a wooden house to be built at the edge of a forest, or is compliance with a tree-fall boundary to be confirmed?

No, completely misunderstood. We are building with a construction company in a new housing development. The development plan is in place. No special requests, just a completely normal city villa. A building notification (paragraph 62) and a simplified procedure (paragraph 63) are two different procedures.

Regarding our previous progress: We submitted documents for a simplified procedure, after 3 weeks we received a letter with many notes on what still had to be supplemented/proven. We went back to the architect, who revised the documents and sent them again. Due to this change, however, we are out of the simplified procedure (statement from the building authority). In addition, the building authority requested a special permit from the municipality because of the aforementioned window. The processing time of the municipality – 2 months.
Of course, we want to avoid this deadline. We called the building authority again and they showed us a possibility of a building notification. The architect advises against it, probably does not want to assume any risks because in this case, you basically build without any approval and are later liable for possible deviations yourself (which in our case basically do not exist). I hope it’s a bit clearer now.

It remains incomprehensible to me, bordering on nonsensical: where a permit exemption and a building notification differ, the latter usually applies to approval-free but notification-required structures, such as individual carports in some federal states, for example.

Additionally, the building authority requested a special permit from the municipality,

I do not understand that at all yet ;-)

We can try to specify that:
Which federal state is it about?
And what does the [Forstamt] have to do with all this?

Apart from being smart alecks (sorry about that), I can contribute a little bit of information. We have just built. New development area with zoning plan in NRW. We (meaning us and the architect) are now jointly liable in the simplified procedure to ensure that our house is exactly as it is allowed to be according to the zoning plan. This concerns many things like eaves height, ridge height, building window, DN, color of the roof covering, etc. We signed the application submitted by the (independent) architect, personally handed over all the paperwork at the municipal building authority, and 3 or 4 days later the approval was already granted.

The municipality is released from liability in the simplified procedure, so usually there are no problems for them ... unless the submitted documents deviate drastically from an approvable building. I have also heard of such cases here where, for example, 4 residential units were to be built on one plot, but only 2 were permitted. And demanding demolition is no fun for either the municipality or the builder.

Except for smart aleck remarks (sorry for that)

One simply cannot also guarantee, with the best intentions, that good advice will please the advisee.