Set up a website on my internal network and access it with a domain name instead of IP

BIND is a DNS server. (The ND means "Name daemon", which is a Unix-ish way of saying "name server".)

You can have BIND tell any device, including the laptop (and including the server), that mynet.home has an "A record" of 192.168.2.3

You seem to have understood all of that (vaguely, at least). Then you'll want to make sure that your computers (the laptop, and the server) send DNS queries to 192.168.2.3

One way to do that is to specify which system to query. e.g.:

nslookup mynet.home 192.168.2.3

A better way is to make the default location for DNS queries to be the server. Rotunduh's comment describes that. The most traditional way for Unix-ish systems is to adjust /etc/resolv.conf and include lines like:

nameserver 192.168.2.3
search mynet.home

Then you can simply use: nslookup mynet.home

You should also be able to ping (unless ping is unsupported, such as if a firewall blocks standard ping traffic). e.g., ping mynet.home

You should then be able to reach your webserver at http://mynet.home

Your web server might not show you the desired web pages until the web server is configured to recognize "mynet.home" as a domain name that it responds to.

So there's a few steps in the process:

• set up DNS server (e.g. BIND)
• set up network configuration to specify where DNS queries go to
• set up web server to support the domain

That's not including testing, or handling any difficulties like firewalls blocking traffic. So I submit this answer as a brief overview, with the expectations that one or more of these steps may be complex enough that you may have additional struggles/questions while setting all this up. Hopefully this roadmap gives you enough guidance to know what direction you're trying to head. (If you have additional specific questions, please make some new questions.)

(Of course, I'm not trying to suggest anything other than following the standard recommended process: check for documentation, online tutorials, etc., and then ask away if things remain unclear.)

Note: I'm not trying to disagree with other answers, which say things like using /etc/hosts and/or adjusting a router's configuration. I'm not trying to say that other approaches, referenced by other answers, won't work. There are multiple possible designs that can work. In fact, I would even say that those other approaches might be the fastest way (in the short term). However, using BIND, as you suggest, is more similar to professional setups like what is common on public Internet servers. And using BIND ought to work fine. You seemed to be asking about that approach, so that is what my guide is trying to describe.