Adding to your last paragraph: Medium to large sized ISPs and web entities are also highly likely to use a third domain, which never touches the public internet. You might have:
a) your public web presence, www.widgets.com for marketing, sales, customer web portal, customer billing, and so forth.
b) your domain name used for public reverse DNS for your ARIN, RIPE, APNIC etc IP space, for an ISP that has its own AS, such as as12345.net. This is the function that the 1e100 domain serves for Google.
c) your internal domain name that is used by your management network to address every piece of network equipment, hypervisor, virtual machine and so forth. This could be something like "widgets.internal". The company internal DNS servers that never touch the public Internet will be authoritative master/slave for this, and your intranet clients will be set up to query these DNS servers. Your reverse DNS for all of your RFC1918 IP space (10.x, 172.16.x, 192.168.x, etc) will have forward/reverse matches for everything that you manage and monitor in your network, so that automated tools can crawl the network and auto discover/auto-provision new equipment.
Internet users outside of your intranet will never see item C, but it definitely exists in a lot of companies' infrastructure.