However, since I am logged in will they know I downloaded it?

i highly doubt it makes a difference if you download or stream. Both is just a request for that file.

Sure you could log the file requests and if it's "transfer -> pause -> transfer" you probably streaming and if it's just one transfer for the whole thing at once it could be a download, but if they can detect that they probably have the transferrate limited anyway.

Also if you can see how much it buffers and it buffers the whole thing at once while listening there should be no difference between a dl and a stream for them to see.

TLDR: don't worry and dl. Just be sure the files have no embedded token or other identifier before you share them. If you don't plan to share just dl.

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It specifically says you are not allowed to take it

Just like every other service out there. Main reason being "if it's not forbidden it's allowed" so they forbid it.