Be very careful here.
1. R’ Chaim is truly a gadol. And I have also heard this b’shem the Bostoner Rebbe. But that does not mean we pasken like this.
2. There is most definitely a difference BEFORE the name is given and AFTER. One thing might be said when asked if one should give the name Shira, and another if one asks “My name is Shira”
3. You could also have issues with things like Kesubas and Gitten.
4. I have mostly heard this before with “nicknames”. When people come and say their name is “Hershel” (Which is really, Zvi, Naftoli, or Naftoli Zvi) or “Leibel” (Yehuda, Aryeh or Yehuda Aryeh). I forget if this is brought down in the Chassam Sofer or the Chozon Ish. I heard it b’shem R’ Frand, then looked it up where he said it was (Yes, it was there)