This is a good question. I think it best always to regard the values of $\sin$, $\cos$, and so on as ratios. What allows these values seemingly to be defined as distances in your quotation from Wikipedia is that that definition refers to the unit circle—a circle whose radius is $1$. A similar definition that works for circles of arbitrary radius would be
The trigonometric functions $\cos$ and $\sin$ are defined, respectively, as the $x$- and $y$-coordinate values of point $A$ divided by the radius of the circle.
In this definition, the values are again ratios.
Added: There is a sense in which lengths stated within some system of measurement are ratios too. To say that a tree is $3$ meters high is to say that the ratio of its height to that of the fundamental unit of measurement—whether that's defined by an actual meter stick somewhere, or something else—is $3$. When we quote units with our lengths, we are implicitly carrying along a physical length to be used for comparison. So mathematically, lengths quoted with units are pairs, where the two elements of the pair are
- the ratio of that which is being measured to the fundamental measuring unit, and
- the fundamental measuring unit itself.
When we talk about the unit circle, we are abstracting away from that a bit by using the circle's radius itself as a measure, rather than something external. All quoted lengths are now really ratios of lengths defined within the figure itself since the unit of measure is within the figure.