Yes that is correct - with cold migration the data does move across the management network and this is the espected behavior -Traffic does not through the vMotion network because a Cold Migration is not vMotion - also depending on whether your shared storage supports VMware Storage APIs or not will depend on whether the data stays on your storage network or onto you public network.
Also 10 GB in my opinion is overkill for your vmotion network - it is designed to work with 1 GB -