How much does a successful Distinguishing Attack "break" the attacked stream cipher?
I would say a distinguishing attack should count as a break. Especially so if it is practical. 

The reason for this is that if you can distinguish the key-stream from random, you invariably leak details about the plain-text.

For example, suppose somebody turned up a few terabyte disks encrypted with VMPC under the same key. It says in the paper that after approximately $2^{40}$ bytes the cipher can be distinguished from random. The bias in the underlying cipher might allow an attacker to work out whether the plain-text was written in Arabic or English.

That in itself is useful information to know about the plain-text and, to me at least, constitutes an unacceptable weakening of the cipher.