I'm glad you enjoyed and found it touching! Thank you so much for playing! I'm glad you managed to play it, despite the trouble! :)
!! SPOILERS !!
So the only person that knows that you changed your name is actually Flowey. He will comment on it and ask why you changed your name. He'll even make a callback to whatever the first name you picked was. But he still knows (or believes) that it's the same person playing.
And the same goes with Sans. Even if you change your name, he still knows that it's you. His SOUL is connected to yours as your SOULmate. And that connection is unique. So having a different name won't make a difference for him.