Is the general equation for a straight line not considered a linear function in linear algebra? Is the general equation for a straight line, which we called a linear function in highschool, i.e.
$$f(x)=mx+c \tag{1}$$
not considered to be a linear function according to the linear algebra definition that it need to satisfy
$$f(\alpha x_{1} + \beta x_{2}) = \alpha f(x_{1}) + \beta f(x_{2}) \tag{2}$$
since the two sides of equation $(2)$ would differ by the constant term if using equation $(1)$ as the definition of $f$? As far as I understand the first equation I posted would be considered an affine function but not a linear function in linear algebra (right?).
Are there just (annoyingly) two completely different definitions of what a linear function is or is there a reason that these two definitions have the same name (i.e. are the concepts related)?

Simply put, the first one refers to "a straight one-dimensional figure having no thickness and extending infinitely in both directions". The second one refers to "verifying the linearity property". Everyday words are polysemic, and the first notion of a line possibly predates the algebraic notion. 
In the past, the first one was callled a "right line" or a "straight line".
Some have called "line" the edge of a graph, in differential geometry the line can be interpreted as a geodesic.
Polysemy exist in mathematics too, and your knowledge and jugdement will help you distinguish the different acceptions. For instance, binary refers to the binary number systems, or to the nature of an operation acting on two operands.
You can read about the polysemy of symbols in: Polysemy of symbols: Signs of ambiguity. 
