I think that is about how to set up a personal API key.
That is not relevant to this problem. Any API key from any account can be used as long as it is authorized to access the data.
What is important in this situation is that the Teamdrive user who has write-access to destination also has read-access to source. Otherwise there is no way for that user to ask the Google server to copy that data. When you do it via your own PC it does not matter if the different places use different accounts without cross-access - but when you server-side transfer it must be this way.
Normally Content Manager should give sufficient permissions to read&write - but I have seen some situations where it actually doesn't work unless you have full Manager access. It is a bit weird - and I'm not sure why this is. Might be a bug - but you should at least be aware of it so you don't get frustrated trying to find out why something doesn't work 
I think Manager is mostly required if you actually move files (ie. delete on source) rather than just copy. Especially if the account that is deleting is not the same that uploaded the data. I will leave it at that