Okay I've been seeing a LOT of confusion with regards to 13 so I broke a few points down so that it's easier to understand. taihaole recommended I added this as a discussion topic after reading this in the comment section. We all know how fast that comment section moves.

When Sun Jae is dying in the water it creates a new timewatch. By the time he is out of the water he has died thus "activating" it to be used by Im Sol. Sun Jae's soul has left his body, so only Sol would have traveled back. The watch takes the person to the time they think they need/want. In the OG timeline she mentions that Sun Jae went to the school next to hers and laments that she should have known him then and why wasn't she paying more attention to him. So when she wants to save him thinking that he took his own life because of being in the entertainment industry it takes her to before he becomes a trainee.
When she's trying to go back again while in 2009 she thinks it's better if he had never met her at all so the watch takes her to before they met for the first time and he falls in love at first sight.

On Dec 7th 2023 Im Sol mentions having been back for a few seasons now and that she's still not fully adjusted to her life without him in it. That means she eventually came back to her time line like before on Jan 1, 2023.

When she is talking with Tae Sung he mentions her meddling in other people's business taking cigarettes and teaching kids a lesson. This means that they meet the same way they did in the first altered timeline. She still went after the kids smoking outside of the gym trying to protect Sun Jae. She was more discrete this time around while they were in high school.

Sun Jae having never met Sol wouldn't have written the song Sudden Shower nor would he have performed with Eclipse at the competition. He only sang with them to win a phone for Sol. He joined the band in the OG timeline as an outlet after Sol's accident/kidnapping...he knew she liked musicians and he needed to get his emotions out. AKA he's never scouted with Eclipse as a singer. Maybe In Hyuk is able to convince him in some other way, but I'm not sure he's anything but an actor in the newest updated present day.

Sun Jae would have remained focused on swimming and blown his shoulder either during the competition or practice...it was meant to happen no matter what. His father would have gotten him to that rehab. As he had no outside interests other than swimming Sun Jae would have gone.  

Because he never met Sol outside of her house he never changes his bus routine to see her and watch her from afar. Even if Sol(young) would have fallen asleep on the bus she would have stopped further down the bus line NOT in the middle of Kim Young Soo's hunting ground.

If Sol(older) remembers that KYS didn't just kidnap her but is already a murderer she could still report the abandoned building from the OG timeline to Tae Sung's father.
Even though she's the only one fully remembering all the timelines she is leaving an impression on people she had previously never interacted with in the OG timeline, aka Tae Sung, Detective Kim, Ryu Sun Jae.

Oh, Week 13 was so good, and thank you  MadelineMaureen for your great explanation.
So, the timewatch functions are based on the user's desires and thoughts about when they need to go to change the course of events. Im Sol's attempts to save Sun Jae lead her to different pivotal moments in their lives, influenced by her thoughts and feelings while using the watch.  Is this correct?

 portia7mcg:

Oh, Week 13 was so good, and thank you  MadelineMaureen for your great explanation.
So, the timewatch functions are based on the user's desires and thoughts about when they need to go to change the course of events. Im Sol's attempts to save Sun Jae lead her to different pivotal moments in their lives, influenced by her thoughts and feelings while using the watch.  Is this correct?

This is my assumption based off of what we have been shown so far. I have no idea what is causing the jumps back to the present as those seem random.(if anyone has any new theories about this please let me know) But the narration from Im Sol around the times of the watches being used, especially the first time both watches are used, makes it the highest probable answer. Unless production out right tells us otherwise that's what I'm personally going with.   

I believe that the watch got it's power from Sun Jae's extreme feelings about/emotion towards Im Sol. I believe she can control the watches because of this. I don't think that the watch would have worked for anyone else even if they were to press the buttons at the correct time. Think of it like their own personal red string of fate constantly pulling them back to each other.  

I agree. Just as spirits of deceased people are said to linger till their matters in the mortal realm are resolved, Sun Jae's intense feelings for as well regret towards Sol seemed to latch onto his watch. 

This answered so many of my questions wow! I was especially curious about how Sol would get to know Taesung if she never tried to protect Sunjae from the kids smoking, but on second thought it does make sense that she would try to protect him without being found out regardless of avoiding him directly. 

 shiawasenohikari:

I agree. Just as spirits of deceased people are said to linger till their matters in the mortal realm are resolved, Sun Jae's intense feelings for as well regret towards Sol seemed to latch onto his watch. 

Yes, his love for her is very strong. 

Lovely work ! GREAT!

Thank you, very much!