According to webster's dictionary, birth is:
"1 a: the emergence of a new individual from the body of its parent b: the act or process of bringing forth young from the womb" <www.merriam-webster.com/dictionary/birth>

Premise 1: The definition of birth as understood by many (along with Webster’s dictionary) is the process of physical birth not spiritual. If one is to talk about the spiritual birth of a matter, one has to specify it.
Premise 2: There is no proof that Gautam Buddha was given "birth" in India by her mother and I don’t believe anyone has argued it so far.

Premise 3: Prince Siddhartha and Gautam Buddha is the same person in flesh and blood. If one was to take the DNA of both these person one would find that they are of the same person. If anyone argues this premise, then they can argue that people are different each passing second and thus have no identity of their own.
Premise 4: So if one is talking about the "process of bringing forth young" in this case Prince Siddhartha aka Gautam Buddha then one can not argue that Prince Siddhartha aka Gautam Buddha was born in Lumbini unless you regard them as two separate being which will be ruled out by the DNA test mentioned earlier.
Conclusion: Gautam Buddha was born in Lumbini.
Any arguments against my premises are welcome.