Hello,
RIPv1 actually DOES summarize, but only to classful boundaries. Have a look at the RIP database below (v1):
R1#sh ip rip database
10.0.0.0/8 auto-summary
10.1.0.0/16 directly connected, GigabitEthernet0/1
192.168.111.0/24 auto-summary
192.168.111.0/24
[1] via 192.168.123.2, 00:00:01, GigabitEthernet0/0
192.168.123.0/24 auto-summary
192.168.123.0/24 directly connected, GigabitEthernet0/0
The problem you have in your network with RIPv1 is that the two 10 networks, 10.1.0.0/16 and 10.2.0.0/16, will not be able to reach each other, because RIPv1 does not support VLSM (Variable Length Subnet Masking), and will only send out the classful 10.0.0.0/8 network. Since that is directly connected on both ends, nothing will be sent at all.
See how the RIP database changes when you enable RIPv2:
R1#sh ip rip database
10.0.0.0/8 auto-summary
10.0.0.0/8
[2] via 192.168.123.2, 00:00:15, GigabitEthernet0/0
10.1.0.0/16 directly connected, GigabitEthernet0/1
10.2.0.0/16
[2] via 192.168.123.2, 00:00:06, GigabitEthernet0/0
192.168.111.0/24 auto-summary
192.168.111.0/24
[1] via 192.168.123.2, 00:00:15, GigabitEthernet0/0
192.168.123.0/24 auto-summary
192.168.123.0/24 directly connected, GigabitEthernet0/0
Now the VLSM mask is being sent, and both 10 networks can reach each other.
Does that make sense ?