It's making sense..what you're doing...so just to double check, the laptop is the only one you want to join the domain?
With active directory...it's all build on top of DNS. Your server(s) is a domain controller. It needs to run on top of DNS, which is why servers that are DCs run DNS themselves..and need to look at themselves for DNS. Clients needs to look at the server(s) for DNS..since they need to communicate with active directory thus have to go through the server, register with the server, and all that DNS ties together. Without that DNS from the domain controller(s)...active directory between the servers, themselves, and the clients...cannot function.
Your router has no idea about your server, or internal DNS...active directory, etc. Plugging in public DNS servers like your ISPs..or Googles...or safe DNS services....any public DNS server, they do not know about your active directory or your server and its DNS....so they cannot be of any use to the client workstations of your domain.
Some people like plugging in public DNS servers like the ISPs for secondary DNS...."in case the domain controller is down or not working". Well...first, a server should not be down. Second...this often leads to broken DNS...if not immediately...it can down the road. Workstations can start failing to register in your internal DNS properly and become orphaned, and domain controllers can do the same...since they stop communicating through that internal DNS. There is a timeout period when a workstation will query the primary DNS...not get an answer quick enough, and immediately query the secondary DNS. A poor performing server can fail to respond to DNS queries fast enough and cause the clients to turn to their secondary DNS....and the breaking of that important DNS communcation begins...typically unknown to the end user. Sure...workstations get internet access and all appears well..but really DNS is breaking internally.
So your servers DNS role needs to be there for both internal resolution (and active directory)...and also respond to queries for public domains (surfing the internet). This is where the DNS forwarders come into play (under DNSMGMT.MSC).
Time for me to take some time to write up a fresh article on doing this with a newer server....may take a few weeks but I'll do one and post it in the articles here.