Chrome: Use Apps, Not Tabs (08/2024 Update)

>I’m not completely sure what distinguishes a website that has an “Install” option from one that doesn’t,[...]

You're talking about PWAs, which require things like a manifest and service worker to make them "installable" by the end user. See e.g. here:

https://web.dev/explore/progressive-web-apps

https://en.wikipedia.org/wiki/Progressive_web_app