sorry, I was on a business trip.
External Network adapter is, if you like to communicate with anything that is on your Hyper-V Server or on any other Server in your network. In 99% of all cases this is the card you want to have.
Internal Network adapter is, if you like that the VM is able to communicate only with other VMs on your Hyper-V Server or the Hyper-V Server itself
Private Network adapter is, is like the internal but only the VMs are able to communicate with each other. The Hyper-V Server will not be reachable
With VMs* With Hypervisor With Network
External yes yes yes
Internal yes yes no
Private yes no no
* = on same Hyper-V Server
In your case both adapters should be external if I understood you correctly. At least this will be the adapter, that has no problems with communicating everywhere.