In SSHHook the timeout argument of the constructor is used to set a connection timeout. This is fine.

But in SSHOperator the timeout argument of the constructor is used for *both* the timeout of the SSHHook *and* the timeout of the command itself (see paramiko's ssh client exec_command use of the timeout parameter). This ambiguous use of the same parameter is very dirty.

I see two ways to clean the behaviour: 

1. Let the SSHHook constructor be the only way to handle the connection timeout (thus, if one wants a specific timeout they should explicitely build a hook to be passed to the operator using the operator's constructor).
2. Split the timeout argument in SSHOperator into two arguments conn_timeout and cmd_timeout for example.

The choice between 1 and 2 depends on how frequently people are supposed to want to change the connection timeout. If it is something very frequent. then go for 2. if not go for 1.

BR and thanks for the code!
airflow/providers/ssh/hooks/ssh.py
if private_key:
                    self.pkey = self._pkey_from_private_key(private_key, passphrase=private_key_passphrase)
                if "timeout" in extra_options:
                    self.timeout = int(extra_options["timeout"], 10)
                if "compress" in extra_options and str(extra_options["compress"]).lower() == 'false':
                    self.compress = False
                host_key = extra_options.get("host_key")
                no_host_key_check = extra_options.get("no_host_key_check")
                if no_host_key_check is not None:
                    no_host_key_check = str(no_host_key_check).lower() == "true"
                    if host_key is not None and no_host_key_check:
                        raise ValueError("Must check host key when provided")
                    self.no_host_key_check = no_host_key_check
                if (
                    "allow_host_key_change" in extra_options
                    and str(extra_options["allow_host_key_change"]).lower() == 'true'
                ):
                    self.allow_host_key_change = True
                if (
                    "look_for_keys" in extra_options
                    and str(extra_options["look_for_keys"]).lower() == 'false'
                ):
                    self.look_for_keys = False
                if host_key is not None:
                    if host_key.startswith("ssh-"):
                        key_type, host_key = host_key.split(None)[:2]
                        key_constructor = self._host_key_mappings[key_type[4:]]
                    else:
                        key_constructor = paramiko.RSAKey
                    decoded_host_key = decodebytes(host_key.encode('utf-8'))