I have created a webmap that I'd like to share only with certain people, not with the public, but I am unsure how to achieve this. I want it to behave like an unlisted youtube video, where only people with the link can view the map, not the general public. I have a public ArcGIS Online account and an ArcGIS license as well. Do I need to purchase rights to send private links to webmaps? And if so, which product do I need to purchase?
You need an ArcGIS Online subscription to share maps privately to other subscription holders. Without one, there isn't any concept of sharing privately with a link, like some video hosting websites allow. You can only share publicly.
If you are already an Esri customer, I suggest you contact your sales manager as you may already have access to ArcGIS Online. If not, you can learn more about subscribing from the ArcGIS Online website. Here's a link that will give you an idea of pricing. ArcGIS Online Pricing | Contributor & Viewer Subscriptions.
Is there a diagram or document explaining how different sharing schemes work with the subscriptions (creator vs. user/viewers/contributors)? I have been trying to find an answer to a similar enquiry and so far your explanation is the only one explains how it works in a nutshell!
Sharing is actually pretty simple. If an organization member can create content (based on their license), then they can share that content to others. Here's a link to the documentation on sharing.
This blog post describes an upcoming change to sharing. https://community.esri.com/community/gis/web-gis/arcgisonline/blog/2019/11/22/whats-coming-in-arcgis... There's a good diagram in here that conceptually shows how sharing works.
Sharing includes who can access an item--you, your organization, or everyone--and what groups it's shared with. "Private" sharing means sharing to specific groups that contain specific members. If you want to be very restrictive and share with only one person, you'd have to create a group where you and that person are the only members, and share your item to the group.
hope this helps,