I suppose if one had a definition of a tangent line, defined for all curves, then one would only need to know which way is vertical. [Thurston, On the Definition of a Tangent-Line (1964)] explores several definitions. A geometric one is the following, where an arc is defined to be the homeomorphic image of a closed, bounded interval.
The line $L$ through a point $C$ of an arc is a geometrical tangent to the arc at $C$ if, given any positive number, there is a circle with centre $C$ such that the angle between the line $PC$ and the line $L$ is less than the given number for every point $P$ (other than $C$) which is both on the arc and in the circle.
This would mean there is a tangent line at a cusp, and, in the case of $y = f(x)$, it is equivalent to def. 1. Thurston points out, more or less, that in the parametric approach one requires that "the ratio $X'(c) \colon Y'(c)$ exists," which it won't at a cusp.
The OP asks about tangent line explicitly in terms of a function and not of curve. It is easy to talk about a nonvertical tangent in terms of the explicit linear function $f(c) + f'(c)\,(x-c)$, provided $f'(c)$ is defined. Thurston calls the corresponding line an "explicit tangent." But it seems to hard me to talk about vertical tangents without considering the graph of the function as a curve in the plane, except strictly in terms a definition like one those given in the OP. Note, too, that the definition of the tangent line as the limit of the secant line leads to def. 1, assuming that the notion of the limiting position of a variable line can be made sufficiently clear.
I guess I have tended to use def. 1 myself, probably because of the textbooks I've used. It's easier for me to adapt to the textbook than for the students
to adapt to my idiosyncrasies, especially first-year students.
But I could live with def. 2.
I don't believe there is consensus
about a standard definition. It's not clear to me how much the parametric situation should influence the single-variable function case. It's fairly clear to me that the distinction between def. 1 and 2 in first-year calculus is not that important (except in each particular class, terminology and instructions have to be clear).
An interesting aside, not particularly relevant to the OP's question, is that while any explicit tangent line $y = f(c) + f'(c)\,(x-c)$ is equivalent to a tangent line according to the parametric definition, there is a parametrization of a curve that has a tangent line at a certain point $C$ but which is not locally equivalent to a graph $y = f(x)$ with an explicit tangent at $C$. The parametrization involves one of my favorite functions and I leave the reader to ponder it (or read Thurston's paper):
$$
\cases{X(t) = t^2 \sin t^{-1} + {1\over2}\,t & if $0<|t|\le1$, and $X(0)=0$ \cr
Y(t) = t\,X(t)\cr}
$$
