Yes. A "natural born citizen," within the meaning of the Constitution, is someone who is deemed a citizen at birth. The other side of that same coin is that anyone who isn't deemed a citizen at birth but, rather, needs to undergo a "naturalization" process, pursuant to whatever specific process federal law may provide at the time of that person's birth -- those laws have changed from time to time -- in order to be considered a "citizen," isn't a "natural born citizen."
There's no more to it than that, no matter how much some whack-jobs may wish to think otherwise, and contort history and the law to get there.
By way of application, were it the case that Barack Obama had not been born in Hawaii -- say, that he was born in Vancouver, Canada; the oft-asserted "Kenya" angle is preposterous but, interestingly, the Vancouver scenario is rather less so -- then, under the federal law applicable in 1961, he would not have been a "natural born citizen." That is because, as of the date of Obama's birth, his mother, Ann Dunham, while born in Kansas, and thus a U.S. citizen, had not been a U.S. citizen for "at least five years after having reached the age of 14" -- Ms. Dunham (born November 29, 1942), was aged only 18 years, eight months, and six days as of the date she gave birth to Barack Obama on August 4, 1961 -- which is what federal law required at the time, in the specific circumstance where, as was the case here, the father of the child was not himself a U.S. citizen (as Barack Obama, Sr. was not).
Thus, had Barack Obama been born in, say, Canada under those circumstances, he would not have been a "natural born citizen" because, per federal law at the time, he would not have been deemed a citizen automatically upon his birth. Ms. Dunham would have had to have followed some further process to have her son thereafter "naturalized."
Strange, and rather quirky, but true.
I definitely agree that there's a gray area in regard to foreign born with foreign parentage. Ted Cruz is another example. Had he been elected president, undoubtedly, he would've faced challenges to his eligibility, challenges that had more merit and would've been taken more seriously than any faced by Obama.
Here's the rub. Who is on the hook to enforce eligibility? Is it state election officials? The Electoral College? SCOTUS? Or the voters themselves?
I also agree that the "born in Kenya" narrative was totally absurd. Why would've Obama's mother fly by herself, as a teenager, from modern, safe Hawaii to deliver her baby in a mud hut in Kenya, and then enter into a conspiracy to make it look as if he was born in Honolulu? Just completely ridiculous.