Middle column, bottom third of the column. "This will not of course, include persons born in the United States who are foreigners, aliens, who belong to the families of ambassadors or foreign ministers accredited to the Government of the United States, but will include every other class of persons."
The sentence is imprecise. It's the written record of what someone said, and sometimes people speak loosely. If you read "aliens, who belong to the families of ambassadors...." as the expansion of "foreigners", then that is correct. If you read each element "foreigners", "aliens", "who belong to the families..." as separate and definitive, then it's not correct.
Fortunately, we don't have to rely on this sentence. It is not law. It's a speech. The actual meaning of "under the jurisdiction" is settled law and it has a known meaning. It does not exclude children of aliens, legal or otherwise.
"Under the jurisdiction" covers anyone who is not a foreign diplomat or invading soldier. In other words, anyone who is subject to the "jurisdiction" of our laws, as diplomats and soldiers are not. Illegal aliens and their children, are.