According to these, James was a Simcoe Loyalist; that is, an American who responded to Simcoe's call for settlers on the Niagara Peninsula, and what is (now incorrectly, but in 1867, correctly) called Southwest Ontario. Some Simcoe Loyalists (and this appears to have been the case here) Actually moved to Upper Canada feeling uncomfortable in the new republic, and more at ease in the British held portion of North America; however many also went for other reasons; particularly the free land. The conclusion, in the second web site, that:
"James is documented as a Loyalist; see Orders in Council in Ontario, Canada dated 12/22/1801. He applied for land grants in Ontario in June 1794"
does not follow from the evidence given. As a rule, most Loyalists went to what is now Canada in 1783; and if James arrived so long afterwards, the land grants do not confirm him as one; rather create doubt that he was one. A common misconception. I would think that, considering the rabid sentiment in Orange Co., NY of the active Patriots, that he would not have been allowed to remain in 1783 if there was the slightest suspicion that he was a Loyalist. He would have been convicted without a trial, his property confiscated, and possibly tortured. Patriots believed in freedom, justice and liberty for all; unless they thought or looked differently.
From the United Empire Loyalist Association of Canada's perspective, a Loyalist is one who served during the American Revolution in a military or civil capacity. I do not see any immediate indication that James Stewart did so; so would not be, himself, a Loyalist. However, I would not doubt that many of his descendants married into Loyalist families. Thus, while James himself might not be considered such, many of his descendants (including yourself) likely have Loyalist ancestors.