A property of the place, not of the probe

Define it carefully and the confusion dissolves:

$$\vec{E} = \frac{\vec{F}}{q}, \qquad \text{so} \qquad \vec{F} = q\vec{E}, \qquad E = \frac{k|q_{\text{source}}|}{r^2} \ \text{for a point charge}.$$

Ask two separate questions about every charge in a diagram: is it making the field, or only feeling it? A test charge does the second. Replace a $1$ nC probe with a $2$ nC probe and the measured force doubles, so $F/q$ is unchanged, and it had better be, because the sources have not been touched.

One more restriction goes with the word test: the probe is kept small enough that it does not disturb the sources. Bring a large charge near a conductor and it rearranges the very charges whose field you were trying to measure.

The magnitude of the force on a charge is $|q|E$, and the direction depends on the sign:

1. Positive charge: force along the field arrow.
2. Negative charge: force opposite the field arrow.

Copying the field arrow onto every charge in the diagram is the error, and it produces an electron that falls the wrong way between plates. The quick check runs through the source: field lines leave a positive plate, and an electron is pulled back toward that plate, not away from it.

Between parallel plates the field is uniform, so the force on a given charge is constant, and the motion is projectile motion with $\vec{E}$ playing the part gravity usually plays. That is why so many of these problems look like kinematics wearing a new hat.

Field lines are a sampling of something that exists everywhere. Three rules cover most of what students get wrong.

1. A line gives the direction of the force, not a trajectory. A charge with any sideways velocity leaves the line immediately, exactly as a thrown ball does not fall straight down. Follow the line only for a charge released from rest in a field that does not bend.
2. Blank space is not zero field. The map draws finitely many lines from a continuum. Read the field at a point from how the nearby lines run and how tightly they are packed. Counting how many lines touch a spot is not a measurement.
3. Density carries strength. Crowded lines mean a strong field. Lines never cross, because the field has one direction at each point.

A genuine zero shows up as a place where the source contributions cancel, such as the midpoint between two equal positive charges, and on a good map the lines pull away from it symmetrically.

The net field at a point is the vector sum of each source's contribution, pointing away from a positive source and toward a negative one. Draw the arrows before adding anything. Two cases that get swapped constantly:

1. Midpoint between two equal positive charges: the two arrows point opposite ways and cancel. $E = 0$.
2. Midpoint between a positive and an equal negative charge: both arrows point the same way, from the positive toward the negative, and they add.

For charges at the corners of a square, resolve each contribution into components, total the horizontal parts and the vertical parts separately, then reassemble.

Finally, the interior. In a conductor at equilibrium, carriers move until nothing is left to push them, so excess charge sits on the surface and $E = 0$ inside the metal. In a charged insulator, charge stays where it was put, throughout the volume, and the field inside is generally not zero. Outside either one, a spherically symmetric charge acts exactly like a point charge at the centre.