I think what confuses is the fact that the definition in the book is just an informal way of saying what you said (modulo the line meeting the curve in 2 points - I'll get to that immediatly): In my expreience a lot of calculus books are deliberately imprecise so as not to frighten the student with technical definition and thus give first a definition which is supposed to appeal to the intuition. We could state the definition of an asymptote at different levels which vary in rigor and formalism. Thus we could have:
Less precise (at an essential point) and informal - your definition: "The asymptote to a curve is a straight line such that the distance between the curve and the line tends to zero as they tend to infinity" (Lack of precision concerning whether you meant only $+\infty$ or not and if not, if you meant "they tend to $+\infty$ or $-\infty$" or they tend to $ +\infty$ and $ -\infty$")
Less precise (at a not so essential point) and informal - The books definition: "An asymptote is a straight line which meets the curve at two coincident points at infinity" (Lack of precision concerning how to rigorously translate "meets the curve" into a mathematical statement)
More precise and semi-formal - Wikipedia's definition. Note that to be formal, one has to distinguish many cases in which the graph of the function could behave. Also note that in that definition arbitrary curves are excluded, since these can be rather monstruous and to ask for an asymptote for such a thing wouldn't make sense; see for example this thing ].
Note further that 1) the more formal you get, the more context you have to specify (which in the other cases is implicitly assumed. Example: You were only talking about curves of graphs of functions on arbitrary curves in $\mathbb{R}^2$) 2) these 3 levels aren't by far the only ones; one could insert many levels of rigor/formalism between these three and there would also by room. 3) definitions aren't carved in stone; different authors use slightly different definitions (which mostly vary in technicalities; the underlying intuition is almost always the same).
To address the last problem (of the first question) concerning the line meeting the curve in 2 points: This is a special case of the note 3) from above: Some authors state their one definition for various reasons. My guess, as to why the author required that the line should meet the curve in 2 points at infinity, is, again, that he wants to remain intuitive - and this definition of an asymptote just does that. The graph of the other answer illustrates this: the line given by all the point in $\mathbb{R}^2$ that satisfy the equation $y=0$ is, in his definition, an asymptote, which would also be clear by just looking at the graph (which accounts for the intuitiveness of the definition). Usually asymptotes are defined like in your definition, where one is only concerned if the line is "close enough" to the function graph either at $t \rightarrow +\infty$ (I have namend the variable of your function $t$) or $t \rightarrow -\infty$ (instead of $t \rightarrow +\infty$ and $t \rightarrow -\infty$ like in the books version), but then you would have examples of functions and lines, where the line is an asymptote to the function, but the resulting picture isn't so nice anymore, since the line would approximate" the function nicely only for, say, $t\rightarrow +\infty$.
After all this it, you should be now aware, that your definition is a little bit to vague , since you haven't specified if you meant with "infinity" only the positive infinity. If you did, the answer is no.