This can be addressed in one of two ways: George Romney (1968 presidential campaign; Mitt's father) was born in Mexico, and yet he ran for president. The ambiguity of the constitutional requirement was theoretically satisfied by the fact that his parents were US citizens and, therefore, their child was a citizen at birth. See this article, for example. Given this, someone born anywhere to a mother who was a citizen would be eligible to run for president, but ultimately, the electorate would be the best guide as to the interpretation of this clause of the Constitution: if the electorate elected someone president born outside the US to a mother who was a citizen, then the interpretation of the Constitutional clause would be satisfied by popular demand.

Your specific question is about a birth in a territory - and this could presumably include those born to parents who were not territorial or US citizens. Barry Goldwater (1964 presidential campaign) was born in the Arizona territory, and yet his qualification to run for president was never challenged. Similarly, John McCain (2008 presidential campaign) was born in the Panama Canal Zone - to US citizens. See this article, for example.

edited to insert years of presidential campaigns.