>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?](https://travel.stackexchange.com/questions/179493/can-i-enter-germany-as-a-tourist-without-a-return-flight-with-my-german-husband/179495#179495)

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.