Your statement "VLAN1 id 13" is a bit confusing. I'm assuming this means that you are using the name VLAN1 for VID 13. This is an odd and confusing naming scheme. It would make more sense from an administration perspective to use VLAN1 for VID 1 and VLAN13 for VID 13.
When using untagged frames between the switches, you are relying on the ingress filter (PVID) to determine what VLAN the traffic will be part of. A port can only have one PVID. This configuration will work for your existing scenario, but it is not scalable.
The ideal configuration would be to tag the VLANs between the switches. In this case, you would use access mode for ports 3 and 4 (switchport access vlan 13) and trunk mode for port 24 (switchport mode trunk) and add VLAN 13 to the trunk (switchport trunk allowed vlan add 13). The summit switch would then use untagged frames for the client ports and tagged frames for the trunk port. The PVID on the client ports should be 13. The PVID on the trunk port should be 1.