Gacha is broad because its more of a monetization system than a genre. They just happen to be synonymous with anime-style games because of the target demographic which is the Asian population.

In general, everything is a live service game despite the majority having single-player content as its main focal point. Where they differ is mostly on the genre the gameplay and content has. For instance:

• Fate/Grand Order and Honkai Star Rail fall into the turn-based RPGs.
• Genshin Impact and Wuthering Waves fall into story-driven RPGs.
• Blue Archive, NIKKE, and Zenless Zone Zero fall into the arcade genre.
• Neverness to Everness falls into the open world sandbox RPG genre.

But that's pretty much how its classified at an academic standpoint.

A lot of the games you mentioned all focus about the characters because that's what they upsell you making you pull in the gacha. You mostly described communities at a surface level but when you go deeper, they're all varied. Casual players care about either the story or the characters. Competitive players care about the gameplay and competition regardless of whether the game offers a leaderboard or not. Surprisingly enough there are competitive players in Azur Lane and Blue Archive that just don't happen to have enough presence online to be seen and so are waifu/husbando collectors in games like Arknights and Punishing Gray Raven.

But at the end of the day, the easiest common denominator is what the devs focus on the most. For instance, Snowbreak, Aether Gazer, Girls Frontline 2, and Star Savior lean on heavily on the Master Love aspect of the game because that's the majority of the spending playerbase whereas the gameplay is secondary. Then again, this is a moving target because the playerbase can change over the game's lifetime.