Instead, I want it to be a part of another, distinct issue type scheme. I can add my new issue type to my new scheme, but can't figure out how to remove this from the default issue type screen. Is this possible
You can't modify the default issue type scheme though. You will have to create a new scheme and associate it with existing projects if you don't want the new issuetype in them.
See https://jira.atlassian.com/browse/JRA-11222 for more discussion on this.
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I recommend you to refer to the documentation here. It explains the steps to add and remove an association to issue type to a screen scheme.
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