It turned out, the problem I am having here has nothing to do with my server configuration, service account or any other account configuration on my remote machine whatsoever.
The services DO work as they should.
It seems that I either set up my script in the wrong way or it is not working properly. As I looked into the execution of the script I saw that at the end this command was created and tried to execute:
"C:\Program Files\IIS\Microsoft Web Deploy V3\msdeploy.exe"
-verb:sync -source:archiveDir="C:\Users\someName\AppData\Local\Temp\DeployDBVariantion00_zip"
-dest:auto,includeAcls='False',ComputerName='some.ip-.address?site=DeployApp/DeployThis',Username=someOtherName,Password=haveAnotherGuess,AuthType='BASIC'
-disableLink:AppPoolExtension
-disableLink:ContentExtension
-disableLink:CertificateExtension
-setParamFile:"C:\Users\someName\AppData\Local\Temp\DeployDBVariantion00_zip\SetParameters.xml"
-skip:objectName=dirPath,absolutePath="_Deploy_"
-skip:objectName=filePath,absolutePath=web\..*\.config
-skip:objectName=dirPath,absolutePath=_Package
-skip:objectName=filePath,absolutePath=.*\.wpp\.targets$
-allowUntrusted
But for this command to work the ComputerName-Parameter needs to include the full address to the target service AND the application name under the IIS site must not be included, so modifying it a bit into this
"C:\Program Files\IIS\Microsoft Web Deploy V3\msdeploy.exe"
-verb:sync
-source:archiveDir="C:\Users\someName\AppData\Local\Temp\DeployDBVariantion00_zip"
-dest:auto,includeAcls='False',ComputerName='https://some.ip.-.address:8172/msdeploy.axd?site=DeployApp',Username=someOtherName,Password=haveAnotherGuess,AuthType='BASIC'
-disableLink:AppPoolExtension
-disableLink:ContentExtension
-disableLink:CertificateExtension
-setParamFile:"C:\Users\someName\AppData\Local\Temp\DeployDBVariantion00_zip\SetParameters.xml"
-skip:objectName=dirPath,absolutePath="_Deploy_"
-skip:objectName=filePath,absolutePath=web\..*\.config
-skip:objectName=dirPath,absolutePath=_Package
-skip:objectName=filePath,absolutePath=.*\.wpp\.targets$
-allowUntrusted
then did the actual job of deploying to my remote machine. I also posted about this in more detail here, because my problem now went from a server configuration problem to a script configuration problem ;)