Hey! Not sure if this is still relevant to you but I figured out what to do for the second true ending.
Basically, besides what is mentioned in this guide, the thing that really decides which ending you get are the different hangouts with Sona.
The one you got happens when you choose to go to a restaurant with her, give her flowers and go dumpster diving because if you choose those options, she won't bond with the monsters and thus keep believing in her twisted morals.
For the second one, you have to go to Grillby's, give her a gift that comes from the heart and go stargazing as she will start understanding the monsters and doubting herself. To get it, you have to choose all of these!
(spoiler) The ending will essentially be everything seeming to turn out well, lots of things being explained until Chara shows up and fucks shit up.
You know you're on the right track if there's new dialogue after Sona and Papyrus make spaghetti, Sona expresses doubt instead of saying all monsters should be humans (at least I think that's replaced), there is a new scene with Asgore, and Sona decides to come inside when Toriel offers to shelter you from the rain.
Hope this helps and have fun!