One learns something every day, like there is a SAML Login History. Can't understand how I haven't noticed that before. Thanks!
And yes, there's error: "Assertion validation error: Fingerprint mismatch"
So it is something with SHA1 fingerprint values.
EDIT:
I added SAML IdP thumbprint for Vision application and it works now. So simple and dumb that it almost hurts.
Thanks!