I’m just watching Sungkyunkwang Scandal and the ML is a very serious person, hardly ever smiles, but I wouldn’t say he is cold. She Was Pretty, When A Snail Falls In Love (chinese drama ), Scarlet Heart Ryeo and You Who Came From The Stars also comes to mind.
Edited to say there is triangle love in She Was Pretty (not so strong ) and Scarlet heart, so ignore those!
Just between lovers definitely doesn't have a cold ML , Kang Doo was so sweet though it's not exactly what you're looking for I highly recommend it
Because this life is our first matches with your description but it does have a little bit of a love triangle ( it affects the story in a very good way so it's not the typical love triangle in every kdrama), the male lead was really cold but he respected the female lead and the drama was both serious and hilarious and the story was indeed beautiful , the female lead though wasn't that strong but she had self respect I highly recommend it too
Hope I helped :)
I also forgot to add A Poem A Day Wich I'm currently watching , I'm actually not sure if he's the ML yet but one of the main characters is so cold and handsome lol but there's going to be a love triangle so pass on this but it's really amazing , give it a shot , the female lead is so endearing too and believe me you won't regret it or maybe it's just my opinion LOL
+1 for Sungkyunkwan Scandal*, She Was Pretty*, When A Snail Falls In Love, You Who Came From The Stars*, and Because This is My First Life. Also:
The Eternal Love (the web drama)
The Princess' Man
Another Miss Oh* (accidentally a jerk)
The Master's Sun*
A Love So Beautiful*
High Society* (also a bit of a jerk)
Queen for Seven Days*
*In these dramas there might be a second male lead or second female lead that likes one of the leads or is their ex, but they don't sway the leads (and in some cases they don't even confess their feelings).
I've left out dramas where there's a significant love triangle and one of the leads is torn.