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Sorry, but you are wrong - I suspect you haven't read Senator Trumbull's complete reply, just the snippet that seems to say what you want. After your snippet

"What do we mean by ‘subject to the jurisdiction of the United States?’ Not owing allegiance to anybody else. That is what it means"

there's several more explanatory statements that you seem to have omitted (Congressional Globe, Senate, 39th Congress, 1st Session Page 2893 of 384):

“Can you sue a Navajo Indian in court? Are they in any sense subject to the complete jurisdiction of the United States? By no means, we make treaties with them, and therefore they are not subject to our jurisdiction. If they were, we wouldn't make treaties with them

and continues on (next column)

It is only those persons who come completely within our jurisdiction, who are subject to our laws that we think of making citizens; and there can be no objection to the proposition that such persons should be citizens.

"Jurisdiction means “subject to our laws.” Diplomats were not “subject to our laws.” Indians were mostly outside the jurisdiction of our laws, because they were held to be under Tribal authority akin to living in a sovereign foreign nation even while within US borders. Are you claiming that a child born in the US of a foreign father (non diplomat), while in the US is not "subject to our laws"?

It is only those persons who come completely within our jurisdiction
Exactly.

Now let's see what the US State Department says about dual nationals like Obama...

"Claims of other countries on dual national U.S. citizens may conflict with U.S. law, and dual nationality may limit U.S. Government efforts to assist citizens abroad. The country where a dual national is located generally has a stronger claim to that person's allegiance. However, dual nationals owe allegiance to both the United States and the foreign country. They are required to obey the laws of both countries. Either country has the right to enforce its laws"

http://travel.state.gov/travel/cis_pa_tw/cis/cis_1753.html

Those born dual nationals, like Obama, are NOT born completely within US jurisdiction. As the US State Department declares, either country to which the dual national owes allegiance has the right to enforce its laws, and dual nationals are required to obey the laws of both countries.