Hello Vadim
There are four ways to distribute a default route in BGP. The first three have a similar effect, two of which you have already described in your post:
network 0.0.0.0default-information originate- redistribution from another protocol
The fourth is the network X.X.X.X default-originate command which works a little differently. It is the same as the default-information originate command, because the default route doesn’t actually have to be in the routing table. However, it is different in that the default route will ONLY be advertised to this specific BGP neighbor and not to all BGP neighbors. The default route will NOTbe installed in the BGP RIB of the router on which you apply this command, so it won’t generally be advertised to all BGP neighbors.
Now the default-information originate command actually DOES place the default route into the RIB so that it does propagate it to all BGP neighbors. It is the network X.X.X.X default-information originate command that allows you to advertise a default route that is NOT in the routing table and is not placed within the RIB, and is sent ONLY to the specified neighbor.
I don’t know the details of your specific topology, but I hope this clarification will bring you a step closer to resolving your issue.
I hope this has been helpful!
Laz