[SalesForce] Validation rule to prevent creating records of certain record type not working

DeveloperName is not the same as Name. It has special restrictions, much like API names (can't start with a number, no spaces or punctuation, etc). Check the Developer Name attribute of your record type. The formula editor won't throw an error, but you'll never match a DeveloperName to a string that contains spaces.

From the documentation:

Required. The unique name of the object in the API. This name can contain only underscores and alphanumeric characters, and must be unique in your organization. It must begin with a letter, not include spaces, not end with an underscore, and not contain two consecutive underscores.

These permissions required for creating and editing validation rules:

To view field validation rules: View Setup and Configuration

To define or change field validation rules: Customize Application

You can find these permissions under system permission of profile or Permission set.

Basically if you have standard profile then you can't remove them also removing these permission you will lost a lot of other setup access too so keep that in mind.