Is leadership just another word for influence
Not necessarily. It depends on the role of the leader and the type of leadership exercised.
Influence is certainly a component of leadership and for political leaders, it is a desired outcome. Many politicians are motivated by a desire to change society and if you can't achieve influence then you will fail to engage people to accept/adopt the changes you want.
A corporate leader - a CEO, for example - also seeks influence (both in relation to staff and whatever direction the CEO and board want the company to go in and in relation to consumers).
Some forms of leadership value compliance rather than influence, though. In the military, an officer may expect troops to comply with an order without being persuaded that the order is correct. The key factor operating here may respect for or giving credence to an authority figure.
It might be most accurate to say that influence is a component of leadership and is one of the ways the effectiveness of a leader can be judged.