Vanity domain
A branded version of a SaaS-provided URL. Often a synonym for custom domain, sometimes used specifically for subdomains of the SaaS host.
A vanity domain is a domain that exists primarily to look better than the default URL. Two flavors:
Customer-owned vanity domain. hello.janesbakery.com resolves to your SaaS. Same thing as a custom domain. The customer owns the registration, you own the certificate and serving.
Provider-owned vanity subdomain. acme.notion.site is the customer's chosen slug on a domain you control. Sometimes called a "vanity URL" or "tenant subdomain." You own the registration; the customer picks the prefix.
Which one your users actually want
Almost always the first. The provider-owned subdomain is the consolation prize when you haven't built custom domain support yet. Once you ship custom domains, demand for vanity subdomains drops to near zero among paying users.
The lift difference
Provider-owned vanity subdomain: cheap. One DNS wildcard record (*.yoursite.com) plus a wildcard SSL cert. New tenant signs up, no new DNS or cert needed.
Customer-owned vanity domain: real engineering. Per-domain DNS routing, per-domain cert issuance, per-domain monitoring. This is the work a custom-domain API does for you.