In the Salsa$20/$ChaCha$20$ ciphers, what is the difference between a counter and a nonce?

There is no difference. They are just different terms for the same thing.

The counter is controlled and incremented by the encryption program. It is incremented for each message. The nonce is set to a random number and never changes. It will be the same value for every message encrypted by the program.

Both the counter and the nonce are controlled and incremented by the encryption program. The counter is set and the incremented for each new message. The nonce is set to a random number at the start of a message, and incremented for each time the PRNG generates a new block of key bits.

The counter starts as a new true random number each time the encryption program starts. It is then incremented each time the PRNG generates a block of key stream bits. The nonce is also a true random number that is generated each time the encryption program starts, however it remains the same while an entire message is encrypted or decrypted.

The counter and the nonce are both generathd and controlled by the encryption program. The counter is reset to zero at the start of each new message and incremented each time the PRNG generates a block of key stream bits. The nonce is incremented when the encryption program starts, and the same nonce is used to encrypt or decrypt an entire message,