I had my first experience with Counterspell in a one shot I ran. I kind of made up the edge cases as I went but afterwards with some thought interpreted the spell like the following.

Counterspell works like this:
1. prereq: you SEE a person casting a spell
2. Without knowing what spell or what level you may Counterspell
3. Another spellcaster (or the original spellcaster) sees the 2nd spellcaster casting a spell as a reaction - even if you are the original caster you can wiggle your pinky and Counterspell the Counterspell
4. this can go on an on for all wizards who have reactions to counterspell creating a virtual chain of reactions being cast
5. everything then gets resolved. if you originally counterspelled at a low level and fail against a higher level spell your counterspell fails and the chain of counterspells is mute. If the counterspell succeeds you resolve the next counterspell down the chain. Regardless of results all those spell slots are spent.

casting where you are hidden or if its a subtle spell prevents step 1. You didnt see it so you cannot counterspell it (hearing it doesnt make a difference) (also you need to be within 60ft)

Is this the correct way to interpret the counterspell rules?

Quote from mburnham1973 >>

I had my first experience with Counterspell in a one shot I ran. I kind of made up the edge cases as I went but afterwards with some thought interpreted the spell like the following.

Counterspell works like this:
1. prereq: you SEE a person casting a spell
2. Without knowing what spell or what level you may Counterspell
3. Another spellcaster (or the original spellcaster) sees the 2nd spellcaster casting a spell as a reaction - even if you are the original caster you can wiggle your pinky and Counterspell the Counterspell
4. this can go on an on for all wizards who have reactions to counterspell creating a virtual chain of reactions being cast
5. everything then gets resolved. if you originally counterspelled at a low level and fail against a higher level spell your counterspell fails and the chain of counterspells is mute. If the counterspell succeeds you resolve the next counterspell down the chain. Regardless of results all those spell slots are spent.

casting where you are hidden or if its a subtle spell prevents step 1. You didnt see it so you cannot counterspell it (hearing it doesnt make a difference) (also you need to be within 60ft)

 

Is this the correct way to interpret the counterspell rules?

 Close.

1. You see a creature within 60 feet of you casting a spell.
2. Correct.
3. Correct.
4. Correct.
5. All spell slots are spent, but you treat the counterspells as a stack.  Suppose 5 counterspellers exist (and 1 original spell, say magic missile).  The 5th counterspeller's counterspell resolves first:
   1. Resolve Counterspell 5.  If it succeeds, 4 does not happen, so proceed to Counterspell 3.  If it fails, proceed to Counterspell 4.
   2. 4 proceeds to 3 or 2.
   3. 3 proceeds to 2 or 1.
   4. 2 proceeds to 1 on failure or the original spell on success.
   5. 1 proceeds to the original spell on failure; on a success, end.
   6. Original spell resolves if possible, then end.

Unnumbered interpretation:

1. You can see a subtle spell with an M component being cast, because you see the M component rather than the S.
2. Whether or not hearing makes a difference gets into some potentially contentious territory because some DMs differ on whether blindsight via hearing qualifies as sight for mechanics that demand something see, but otherwise correct.
3. You are correct that 60 feet is always mandatory - increasing the spell's range does nothing, just as it does nothing for Mage Hand.