The definition of Electric field? In Raymond Serway's physics textbook, the definition of the electric field vector, was that it's force vector acts on a positive test charge, given as force divided by the test charge:
$$\overrightarrow{E}=\frac{\overrightarrow{F}}{q_0} \tag{1}$$
Then he provided another equation which is:
$$q\overrightarrow{E}=\overrightarrow{F} \tag{2}$$
where $q$ is any charge, no matter if it is positive or negative.
My question is: the book defined the electric field as the force acting on a unit positive test charge, so where has equation $(2)$ come from? We should say that:
$$\overrightarrow{E}=\frac{\overrightarrow{F}}{q}$$
not
$$\overrightarrow{E}=\frac{\overrightarrow{F}}{q_{0}} \ \ ?$$
 A: Good question. You say $q_0$ is a unit positive test charge. The actual definition of electric field involves the limit $F/q_0$ as $q_0 \rightarrow 0$. The reason for the limit is that we don't want $q_0$ to be large enough to change the charge distribution that is producing the field. As we approach the limit, we find that the ratio $F/q_0$ remains more or less constant so indeed $F=qE$ for another charge $q$. But $q$ must be small enough, so as not to change the charge configuration that is producing the field, or means are provided to maintain that charge distribution, e.g. by maintaining a potential difference.
A: The first formula is a definition. This is how we define the electric field.
The second formula is the effect of the field. It's we put a negative charge in a field pointing to the left, it will experience a force to the right. This is why we call it a "negative" charge: because it reacts to fields in the opposite way from a positive charge.

we should say that $\overrightarrow{E}=\frac{\overrightarrow{F}}{q}$

If we did this, we wouldn't have a unique definition of the electric field. If we tested with a positive charge we'd think the field points one way and if we tested with a negative charge we'd think the field points the opposite way. If someone asked you, "what's the electric field at this point?", you could only answer "I don't know, it depends what charge you are going to place there.", and this wouldn't be very helpful for writing down a single vector that can predict how any charge will be affected when placed in the field.
A: The precise definition can be written in many ways. Your question seems to focus on the definition based on Lorentz force, in other words when the system (the object on which the force is applied) is a point-like charge (with the additional assumption that it's at rest, or that there's no magnetic field).
Let $q$ be this point-like charge. Start with the experimental observation that the Lorentz force is proportional to $q$. This can be written:
$$\frac{\vec{F}}{q}\text{ doesn't depend on $q$}$$
Therefore it can be useful to study this ratio, because it's independent of the system on which the force is applied. Let's call this ratio the electrical field $\vec{E}$.
So far this definition is very limited, because it was built on the special case of a point-like charge. For a system with a finite size, this definition is awkward, so physicists decided to replace it with something else that doesn't require mentioning anything about the system: Maxwell's equations.
Summary: defining the electric field as $\vec{F}/q$ is mostly an introduction to show that it's possible to decouple the source and the object.
A: $$\vec F=\frac 1{4\pi\epsilon_0}\frac {qq'}{r^2}\hat r$$
The problem with the expression of force is that it depends on external factors, that is it is not a fundamental aspect of the charge itself. However, if we remove charge $q'$, we get the expression $q\hat r/4\pi r^2$, where $r$ is the displacement between $q$ and any point in space. This relation is fundamental to the charge itself, independent of any charge other than itself. This expression is the electric field of a single point charge.
The definition that you mention is related to the experimental difficulties of measuring an electric field whereas this one tries to establish why we need electric field at first place.
