Let me be very frank here. For a beginner who is studying limits for the first time (meaning age around 16 years) the terms "sufficiently close" and "arbitrarily close" are very difficult to handle.
This is primarily because such a student at his stage of learning is acquainted mainly with algebraic simplifications/manipulations and the main focus in algebra is the operations $+,-,\times,/,=$. Thus most of the mathematical study is based on establishing equality between two expressions (via rules of common arithmetic operations).
In Calculus the symbols $+,-,\times,/,=$ take backstage and the focus shifts entirely to inequalities (but this point is never emphasized in any calculus textbook). Calculus or analysis is fundamentally based on the order relations like $<, >$ instead of arithmetical operations. Here we are not so much concerned about whether $a = b$ or not, but rather how much near / close $a$ is to $b$ when we already know that $a \neq b$. A measure of this nearness/closeness is given by the expression $|a - b|$ (which is something easily handled by students trained in algebra).
The next issue is that students fail to comprehend the significance of the fact that there is no smallest positive rational / real number (although students know this fact and can supply the proof very easily). Because of this fact we know that if $a \neq b$ then the expression $|a - b|$ can take whatever small positive value based on specific choice of $a, b$. Thus we can choose two distinct numbers $a$ and $b$ which are as close to each other as we please.
Calculus / analysis builds up on such phrases as close to ... as we please and introduces the terms like sufficiently close and arbitrarily close and for this purpose the very powerful notion of functional dependence is used. Thus let the numbers $a, b$ in the previous paragraph have a function dependency on some other variable. To simplify things let $a$ depend on another number $x$ via function relation $a = f(x)$ and let us keep $b$ as fixed. Thus we have a way to choose different values of $a$ by changing the value of $x$.
And then we pose the question : How close is the value $a = f(x)$ to $b$ when the value of $x$ is close to some specific fixed number $c$? Thus we are interested in figuring out how small the difference $|f(x) - b|$ is based on the difference $|x - c|$. If $|f(x) - b|$ is small when $|x - c|$ is small then we say that limit of $f(x)$ is $b$ as $x \to c$.
However to make things precise the smallness of $|f(x) - b|$ and $|x - c|$ needs to be quantified properly and when defining the concept of limit it is essential that it should be possible to make the quantity $|f(x) - b|$ as small as we please by choosing $|x - c|$ to be as small as needed. Thus the goal is to make $|f(x) - b|$ as small as we please and for this making $|x - c|$ as small as needed is a means to achieve that goal. Since the goal is primarily based on our wish (as small as we please) we say that $|f(x) - b|$ should be arbitrarily small (because our wishes are arbitrary and there is no end to supply of numbers as small as we please, remember there is no smallest positive number). And then once we have fixed our goal (say with some arbitrary small number $\epsilon$) we need to now choose $|x - c|$ small enough (or we say sufficiently small and quantify it with another small number $\delta$) to fulfill that goal.
And the next step is the formalism of greek symbols: A function $f$ defined in a certain neighborhood of $c$ (but not necessarily at $c$) is said to have limit $b$ as $x$ tends to $c$, written symbolically as $\lim\limits_{x \to c}f(x) = b$, if for any arbitrarily chosen number $\epsilon > 0$ we can find a number $\delta > 0$ such that $$|f(x) - b| < \epsilon$$ whenever $0 < |x - c| < \delta$.
Instantaneous rate of change of a differentiable function at a point is by definition the change in value of the function when the point is infinitesimally perturbed. The definition of instantaneous velocity at any point itself is the rate of change of position at that point, and is the velocity "at that point".
If $t$ denotes time and $f(t)$ denotes position at time $t$, then the velocity at time $t_0$ is defined as
$$v_0 = f'(t_0)=\lim_{t\to t_0} \frac{f(t) - f(t_0)}{t-t_0}$$
As you mention, the limit is an "actual value", and may be defined based on $f$. When it is defined, it gives the exact velocity at time $t_0$. It is true that when you substitute $t_0$ in the limit, you get $\frac{0}{0}$, which is undefined. But the fact that a function is undefined at a point does not mean that the limit on approaching the point is undefined (take, for example, $\lim_\limits{x\to 2}\frac{x^2-4}{x-2}$). Moreover, substituting $t_0$ represents no change in time, whereas the velocity is defined for an infinitesimal change in time, for which the change in position is given by the limit, and is often well defined.
The key is that taking the limit allows you to exactly compute the change of the position for an infinitesimal change in time. Substituting $t_0$ represents no change in time, and substituting any non-infinitesimal change in $t_0$, say, $t_0+t_1$ gives the average velocity over the time period $t_1$.
Best Answer
To some extent these are a matter of interpretation or even opinion, but I'll give you my take.
In the development you are reading, the tangent line is determined by the gradient at one point, and the approximation through the average gradient between two points is made to come up with the definition. On the other hand, over the years, many people have thought of it as a an infinitesimal change in $y$ divided by an infinitesimal change in $x$, and you are free to do so if this is more intuitive for you. This approach was made rigorous back in the $1960\text{'s}$ by Abraham Robinson.
I would agree with this, at least in a practical sense. How could the velocity at single instant ever be measured?
The instantaneous speed is, I think an approximation, for the reason I gave in 2. On the other hand, the tangent line is a mathematical construct, with a rigorous definition, so I wouldn't call it an approximation.