Not a mathematician but every use of the term algebra I have seen has some way related back to what I learned in grade school about sets of real numbers, operations, properties, and how to solve equations. It's the same thing with linear algebra, they just put restrictions on the kinds of equations to solve. Then more ... abstract there is abstract algebra that goes beyond real numbers and looks at other kinds of sets & operations and classifies them as fields, rings, and more. I wonder if there are examples of the term "algebra" that are unrelated to this. If not that might be one difference between algebra and "a calculus" which as you said seems to refer to any system for formal calculation/reasoning.