I've found what was the problem: in my SendGrid account there was Link Branding enabled. But that's little bit strange because it was enabled before moving to the new server and it worked fine.
Is there a way to remove logo entirely from my email templates
As we can edit the templates, we should be able to remove the logo part. Can you tell me which part of the templates I would need to edit for this?
@maroonbee have you tried experimenting? It shouldn't be too hard to find