In Mitsuha's timeline, Taki still doesn't know Mitsuha yet, if he wrote his name, it would mean that he's letting Mitsuha get hurt by him for a second time.

I love you Taki

After the incident

You're Taki, right? You wrote your name on my hand, didn't you?

Thanks for helping me out on saving my town, and that you said you loved me.

I really don't recall doing that, did someone play a prank on you or something?

Must've been some guy writing on my hand to make a fool out of me

Oh yeah! Isn't this guy the same person I met when I went to Tokyo and mistook him for someone else? I was so upset about it I even had a haircut too.

Waaah... Mistaking the same guy twice, and on top of that I even showed him the words on my hand asking him if he loved me, gosh this is so embarrassiiiiiing!!

8 years later

I feel like I've been searching for something, is it someone? Or perhaps some scenery?

Or maybe I'm just looking for a job? I don't understand myself...

This story also tells us that just a few seconds' lag can lead to a 5-year delay. Every minute and second is precious, so we must grasp all the time we have firmly

Until his meeting with Mitsuha, Taki spent the entire time regretting about making Mitsuha a miserable fool in Tokyo. If the previous part is about saving Itomori, then this is the part of atoning Mitsuha.

I love you

In order to save Mitsuha from the tragedy of seeking Taki via the name on her hand after losing her memory, only to be left heart-broken since Taki doesn't recognize her. Taki figured that he can refrain from writing his own name, instead, by letting Mitsuha write down her name, he can seek out for her own his own.

However this almost become a regret of a life time.

If this is purely a romantic gesture, then Taki wouldn't panic and cry after he found out that Mitsuha didn't write down her name. Before this part we found out how Mitsuha's Tokyo experience went, and in his regret Taki figured out the only logical solution. He is the man who can figure out the entire Operation Itomori in merely a morning, but time played a prank on him.