Is a Mode C or S Transponder a type of Mode 3 Transponder?
Modes 3/A and C are very similar, but the interrogation pulses are timed differently and the responses mean different things.

If you send a mode 3/A pulse, you get a mode 3/A reply (or nothing). The response is a 12-bit number called the squawk code, which is conventionally written as four octal digits, plus an optional ident bit.

If you send a mode C pulse, you get a mode C reply (or nothing). The response is a 12-bit redundant encoding of the pressure altitude between -1200 and +126700 feet, in 100 foot increments.

Note that mode C does not include a squawk code; it is only the altitude. In practice, though, every mode C radar and transponder is also capable of mode 3/A. Likewise, every "mode C" radar always sends both types of pulses and internally combines the two responses, so it looks like a single system. (TCAS only sends mode C pulses since it doesn't care about squawk codes.)

Mode S is almost a completely different system. It includes selective addressing, data frames, message types, and much longer messages (56 bits). It also uses a different, more efficient encoding to allow moving all that extra data without losing efficiency. Of course, for backward compatibility, all mode S transponders can also do mode 3/A and C using the old 12-bit system if needed. Likewise, Mode S radars also send mode 3/A and C pulses, but they're modified very slightly so that mode S transponders will know to ignore them.

1090 Extended Squitter is a relatively minor extension of mode S. A "squitter" means a transponder that sends replies without being interrogated, and "extended" means it added some new message types with 112 bits per message (to hold ADSB data) instead of the previous 56. That's it.