Although I don't have too much experience in setting up a lot of vulnerable servers, "secure" might be a bit vague depending on how you set up your router. Correct me if needed as well.
I would like to have internet connection inside Kali. Usually I would use NAT, but if I also have another connection to vulnerable machines using Internal network, is it secure?
From what I read up on the difference between Bridged and Internal, they're not too different except internal is safer to use than bridged. From what I've read on the VirtualBox manual:
In bridged networking mode, all traffic goes through a physical interface of the host system. It is therefore possible to attach a packet sniffer such as Wireshark to the host interface and log all traffic that goes over it. If, for any reason, you prefer two or more VMs on the same machine to communicate privately, hiding their data from both the host system and the user, bridged networking therefore is not an option.
Source: https://www.virtualbox.org/manual/ch06.html#network_internal
Bridged
From my point of understanding, if you have 2 VMs set up - 1 vulnerable, 1 Kali,
Bridged connection will typically make the connection between those 2 VMs a lot easier due to both being hosted on the same subnet, meaning that if you have an external computer/laptop that connects it to the same router, they can see some of the services/ports that are opened from the IP address of your host computer (That is, if the ports are port-fowarded). The only thing I would worry about here is if there is a service in an open port that you opened and someone connected to your Wifi (or you setup a VPN client/server on your host computer and shared a client packet somewhere), where he/she exploits an zero-day guest-to-host exploit and take control of your host computer, which is extremely unlikely, but possible.
Internal
Internal works the same way, as bridged, however, the only source of communication it has is with the other VM that is set up, where neither VM or the host can communicate with each other.
Overall, internal is a lot more secure if you only want the VMs to interact with each other. Bridged is more towards if you want any other device to interact with the host & the VMs.