I suspect that behind this question lies the notion that an angle should look like some sort of wedge shape, created by two distinct lines meeting at a point, whereas a 'zero angle' just looks like part of a line.
In fact, that was very much the view taken by Euclid in Euclid's Elements, his seminal work of classical geometry, in which he states
"...a plane angle is the inclination of the lines to one another, when two lines in a plane meet one another, and are not lying in a straight-line."
This definition appears to rule out a zero angle, and indeed a 180-degree angle — in fact, the description of an angle by any number of degrees assumes a concept of angle measurement that Euclid avoided: instead he compared their size by mapping them onto each other, or added them by mapping them adjacently to one another, and was thus able to make remarkable progress.
The idea of measuring angles has been of practical interest for thousands of years for purposes such as astronomy, navigation and telling the time, but the use of numerical measurements as a tool for studying theoretical geometry didn't really take off until the work of Descartes and Fermat in the 17th century. Their work led to the study of geometry through algebraic functions and number, and the development of a more numerical understanding of angle was a natural part of that process.
In short, if you think of angles according to numerically-measured size, it is convenient to have a definition that extends along the number line, but if you're thinking of them from a purely spatial perspective, then such an extended definition is perhaps unnecessary and potentially confusing.
The units on a measurement tell you, among other things, how the number will change if you decide to change the units you use.
Suppose that "$1$ unit" in your case is $1$ meter, and you measure the arc length $L$ to be $0.7$ meters. Reporting the angle as "$0.7$ meters" suggests that if we were using centimeters instead, the angle would be $70$ centimeters.
But that's not what actually happens. If we make "$1$ unit" be $1$ centimeter, then the arc will become $100$ times smaller, and the arc length $L$ will change to $0.7$ centimeters.
So it make sense to report the angle simply as "$0.7$" in both cases, because if we change the units we're using, the angle does not change.
Best Answer
" For instance, could I denote an angle like below to denote a clockwise direction instead of the second picture?", yes, you can, providing you show the angle of orientation. Such angles are called oriented angles.