Direct to Reported Speech - a scenario of ambiguity

The question is:

How would one convert

He said, "My name is Harry"

to reported speech?

Rules do say that the converted sentence reads

He said that his name was Harry.

The factor causing discomfort is the 'was' part of it. So it means that his name 'was' Harry and nothing is said about what his name 'is'! So there is a radical change in meaning!! In that light, the converted sentence should read

He said that his name is Harry.

Isn't the latter the correct conversion?

Answer

Having looked at the site that @bigbadonk420 referred to in his answer, I disagree with that answer and interpretation of the quoted site. On my reading of that site, the two options for reported speech are (using the OP's quotation):

with backshift: He said that his name was Harry
without backshift: He says that his name is Harry

There is no suggestion that you can mix the two, which suggests that

He said that his name is Harry

is incorrect, or at least not normal usage, as also confirmed by the first Ngram mentioned by @PeterShore