So, RAW here is how a round with a spell is supposed to work:

1. Character A announces that they are casting a spell. The name of the spell, and other information such as its level are not mentioned.
2. There is a short pause to allow someone to use a Reaction.
3. If no one uses a Reaction, Character A either rolls or tells people what they need to roll.
   • Side note: This is where someone could technically cheat by changing their spell slot level, and is one of the many reasons why Counterspell is a terribly-designed spell.
4. After this roll (and any effect that would apply to those rolls), Character A describes the effect and can optionally state what the spell was:

You all take 36 Fire damage, as an explosion of flame blooms at your feet from my character's 6th-level Fireball.

RAW, Counterspell would occur during that second step. The creature that casts it has no idea what the spell they're countering is, beyond context clues (i.e. they've seen that armored spell caster has been casting spells that heal their allies earlier).

As you said, there are rules to identify a spell. They were added by either Tasha's Cauldron of Everything or Xanathar's Guide to Everything. A character can use their Reaction to identify the spell. This usually means you'll need 2 spellcasters working in conjunction for Counterspell to work with an identified spell.

As for how I run it at my table: I don't. I really don't like it. It's anti-fun, and the awkward pause and wording that's required to cast spells in case someone wants to counter it. There's some equally awkward metagaming thats required if someone accidentally blurts out the name of the spell, and it plays really poorly with how most VTT software handles spells (most just spit them out in the chat for everyone to see). It is just so un-fun that I just ban the spell outright at my table and it makes everything much simpler.

Having said that, if I do play at a table where it isn't banned, we usually go about it as I described above. The Reaction needed to identify the spell is an intentional design decision to prevent spellcasters from identifying every spell cast their way before deciding to counter them, and needing 2 spellcasters to work together to "cleanly" cast it is perfectly fine, in my opinion. Spellcasters are already bonkers in this game, there's no reason to empower them further by letting them save-up their Counterspells until they're absolutely critical.

It's just important that every player is on the same page and doesn't blurt-out their spell names whenever they cast a spell.