As a US citizen you can apply for a residence permit directly in Germany based on §41(1) AufenthG.

See answer where §41(1) AufenthG is quoted:

• Can I enter Germany as a tourist without a return flight with my German husband and kids and apply for family reunion visa there?

You can apply for any type of residence permit.

The 90 day rule does not apply for the time in country X once the application for their residence permit has been made.

Once you leave that country, it will start again.

Since you don't require a visa as a US citizen you can go to Germany on your passport alone (take the proof of residence permit application for country X with you, which is proof that you were in country X based on their national law).

Apply for the German residence permit on arrival.

according to the German embassy, because they said I am NOT considered a resident until I have the actual residence permit, until then I am considered a visitor and must apply from my home country.

The Embassy is correct that you cannot apply for a national visa in country X until you are a resident of country X (i.e. while the application is pending).

But you do not need a national visa (Type D) for Germany due to §41(1).

They also told me that if I went directly to Germany, they would not give me a residence permit while in Germany; I would have to return to my home country to retrieve it.

That information is not correct for citizens of the countries listed in §41.

Yes. I'm a US citizen

As a US citizen you can apply for a residence permit directly in Germany based on §41(1) AufenthV.

See answer where §41(1) AufenthV is quoted:

• Can I enter Germany as a tourist without a return flight with my German husband and kids and apply for family reunion visa there?

You can apply for any type of residence permit.

The 90 day rule does not apply for the time in country X once the application for their residence permit has been made.

Once you leave that country, it will start again.

Since you don't require a Schengen visa (Type C) as a US citizen you can go to Germany on your passport alone (take the proof of residence permit application for country X with you, which is proof that you were in country X based on their national law).

Apply for the German residence permit on arrival.

according to the German embassy, because they said I am NOT considered a resident until I have the actual residence permit, until then I am considered a visitor and must apply from my home country.

The Embassy is correct that you cannot apply for a national visa in country X until you are a resident of country X (i.e. while the application is pending).

But you do not need a national visa (Type D) for Germany due to §41(1) AufenthV.

They also told me that if I went directly to Germany, they would not give me a residence permit while in Germany; I would have to return to my home country to retrieve it.

That information is not correct for citizens of the countries listed in §41 AufenthV.