You know, that's something I don't usually think about that much. It's also extremely subjective. I really like your categories but now that I consider them, I find myself contradicting them with some shows. Using one of your examples, I'm strangely more attached to Chika in Love is War than Kaguya. Kaguya gets the most development (so far in the one season I've watched) but Chika just feels like someone I would get along with more than the other characters. So likability is a factor and a pretty subjective one. No doubt there's someone out there who likes Yoko more than Nia too, though that could turn into its own discussion.
This also made me think about a bit of a weird case, Food Wars (spoilers). Now, the only one of your cases that applies directly here would be the first one, with Erina. She has a relationship with the protagonist (if a hostile one at first), she grows a lot as a character, and it's difficult not to respect her by the end. But when it comes to the title of best girl I would actually say it's Megumi. She arguably has more growth than Erina, going from an insecure mess who's already accepted her fate of being expelled, to a confident chef and one of the most respected students in and out of the academy. However the series doesn't push her and the protagonist together (well, it does in one scene, but it's quickly disregarded and they develop a healthy friendship instead), and despite having probably the most growth out of the entire cast she's not the protagonist. So that would be my subcategory. The surprisingly non-romantically involved side-character who just happens to have (again, arguably) a better arc than the main girl.
Another interesting case would be Monogatari Series (spoilers again). This one has a main character, and he's in a relationship since very early in the series. But these are technicalities because thanks to the nature of the show, we can spend entire arcs without seeing the protagonist and every character can become the main one even if for a little while. So everyone gets a chance. I could give you reasons why almost every character in the show is the best one. Though, I guess for me it does fall into the first category. Best girl would be either Tsubasa, Hitagi, or Shinobu, at least in terms of development. Two of those get some kind of romantic involvement with the protagonist and the other one, well there's no romance, but you could say their bond is more complex than that.
One went from a very traumatizing childhood to gain a very much needed sense of independency that lets her live her own life instead of running away from it. The second one goes through the pain of a torn family, accepts and treasures the negative feelings that come from it and continues to live happily the way she can. And the last one rediscovers her human side after centuries as a vampire and goes from seeking death to accepting and enjoying her life with the MC. "I will never forgive you for what you did, and you will never forgive me for what I did" and "If you were to die tomorrow, I would be ready to end my own life tomorrow—and if you care to live for today, then so will I" are the reason I think Koyomi and Shinobu's is the best relationship I've seen in fiction. Now I don't know what kind of relationship that is, it's not a friendship and it's certainly not a romance, but it's great. Yeah, let's go with that. I'd give the title to Shinobu.
Can't miss a chance to mention Your Lie in April. A nice and simple case where the protagonist is the best boy because of the main girl, and the main girl is best girl because of the main boy. They complement each other so well, won't spoil that one.
Special mention to Rebecca from Lupin III Part 4 because I have her arc fresh in my mind and it was amazing.
For the second category you certainly nailed it with Vin. Almost teared up when she started using Kelsier's phrases in HoA.
I would add Ordinary Person from Akudama Drive to that (yeah, that's the show calls her. She also goes by Swindler). That show is so short. Way to go from fish-out-of-the-water protagonist to I-respect-this-person-so-much-they're-my-new-role-model protagonist in just 12 episodes.
Chihaya from Chihayafuru. She's great, has good development, and we get 75 episodes out of the deal, what's there not to love? This is the kind of show that I watch to give me motivation, it makes me want to rediscover mi passions. And Chihaya is such a good example of my end goal. I can come across as lazy, in more than one way. I don't go out a lot, I don't speak much. I'm scared to tell people the passions and dreams I have because the obvious answer is "Where's your proof?". I say I want to be a game developer but what have I actually done to achieve it? In that regard, characters like Chihaya feel like heroes of sorts. Your Lie in April actually does that too, to an even stronger degree.
Now, that show involves a love triangle and now I'm kind of thinking about the subject. This will be on-topic, I promise:
See, remember Nisekoi? That manga known for rewriting its own premise every five chapters or so? It starts as a love triangle. Now, by the end the story makes fun of itself, joking about the protagonist "eight-timing", but every reader knows only Chitoge and Kosaki, the first two, actually have a chance. Parallel to that, they're also the only ones the "waifu wars" cared about. Sure, the other girls had fans of their own, you could see appreciation comments, but if you found a long heated argument defending one girl over another it was most likely about Chitoge and Kosaki. So I was thinking, maybe that affects our perception? Like we know deep down that we'll be disappointed if we support one of the others and therefore unknowingly force ourselves to pick someone who's actually at the forefront? Or maybe the author just gave them better development because they were the main ones and the others had less fans because of the lesser development? Idk, now I'll have to pay more attention to this.
Anyway, second category, yeah. Let's wrap it with Estelle from Trails in the Sky. It's a game but it's a weeb game, it counts. She's your classical coming of age story but I think it was really well done. She's my favorite game protagonist now. She starts cheerful and immature. She then discovers there's ugliness in the world but she stays cheerful. Seriously, the moment that turned her into my favorite protagonist was one where I was completely sure she would have a breakdown but instead she got up, smiled, and kept going. Such a good arc.
For your last category, I'm so glad to know I'm not the only person who wants to adopt characters. Here's a repeat, Trails in the Sky again but the character this time is Renne. Now, where Estelle is my favorite game protagonist, Renne is my favorite game character as a whole, so there's that. One of the best redemption arcs out there across 3 games, a well told dark past (more writers should learn that it's not only about how messed up you make their past, but about how you tell it. They forget the "telling" in storytelling. In that regard, Renne's is masterful), and she's just really good at making the player want her to be happy. Seriously, that became my end goal for this series, seeing her live happily.
A recent one, Ami from Lupin III Part 5. I haven't finished this but she's playing with the paternal instinct I shouldn't have yet. A girl isolated from the real world, which she only knows through the internet, just now discovering how good and how bad life can be? I just want to show her the good points about the world, she's got plenty of the bad already.
And I should leave it at that. I could probably go through every entry in my MAL page thinking about this subject but that's probably a bad idea. Good topic though, I like when I have to think to myself "Yeah, what do I think about this?".
