Inside a Sandbox, it will be listed as whatever the Organization Edition your production org is. This is because it has the same basic licenses as your production org. You would say that they have the exact same features enabled, except that they are limited in a few ways (e.g. emails sent will have the subject prefixed with "SANDBOX:").
Inside your production org, the Sandbox can be created as several different types, including Developer, Developer Pro, Partial Copy, and Full Copy. This describes how much data storage and copied data the Sandbox has. This isn't the "edition" of the Sandbox, but rather the type of the Sandbox.
Developer is the smallest version with the least storage, Developer Pro gets more storage, Partial Copy gets the full data storage of production, but only copies part of the data, and Full Copy gets the full data storage of production and copies every record from production.
The smallest Sandbox Orgs are limited in space, but can be refreshed daily, and occur quickly (minutes to hours), while the largest orgs can only be refreshed once every 30 days and can take hours to days to refresh. They're also radically more expensive, since they consume as much space as a full production org.
Don't confuse a "Sandbox Developer" org with a "Developer Edition" org. That's a completely different concept. A DE org is used by developers to make packages typically destined for the AppExchange, or learning how to use Salesforce on Trailhead, and or just generally having a separate org for testing or registering a namespace. These orgs are standalone, separate from any production org, Sandbox, or Scratch Org.
Developer Edition orgs also some in several versions, including Developer Edition, Partner Developer Edition, Partner Professional Edition, and Partner Enterprise Edition, which have varying data storage limits, license limits, and features enabled. You need to be a Salesforce Partner to access most of the higher-featured Developer Edition orgs.
Note that you don't use Scratch Orgs with Sandboxes. Scratch Orgs are a special type of Sandbox created through a Dev Hub Org, which may be a Developer Edition (not Sandbox Developer!) or a paid production org (not in a Trial status). Sandboxes can't be Dev Hub orgs.
Scratch Orgs are not in any way associated with a specific Sandbox Org of any type. Scratch Orgs also have a maximum durability of 30 days, after which they are deleted. This encourages code repository-based development, rather than maintaining the "source of truth" in any one org.
