I have been looking for John Adamson parents for almost thirty years, Join the chase, however I don't think the answer will be found on the internet. Old fashioned search is what is needed.
The various spellings have come by numerous sources.
Quakers record disclose- SKUSE and SKUCE. Roberts book suggest SKEW. Other guess have come from speculation. SKUTE came from unsubstantiated posting on gen forum. See History in article "Ann Skute Myth" on AdamsonAncestry.com. This (1998) is the first occurrence of the name "Skute" on any postings (And it was incorrect). You will not find the name used prior to this date.
You will find that many Adamson researchers have taken the Skute misrepresentation as fact. Its too easy to use "I found it on the internet" as a source of ancestry, or link it to others miscomprehension. The bad part is that now Both Ancestry.com and familysearch.org (LDS sponsered site) use these bad posting as references. Users need to read the source disclaimers to see that the source is just another posting.
The source document for the marriage of John Adamson and the origin of the name Ann "Skuse" and "Skuce" can be found on www.AdamsonAncestry.com. Look under Pioneering Spirit of Colonist John Adamson and the Haddonfield article.