What do you mean by "check the test light"?

Many times people are fooled because they don't check (test) the connections to their test gear. When the test light is clipped to ground, it is a good practice to touch the other connection to a known good power connection such as the battery + or the solenoid battery connection. This will cause the test light to light if the ground connection chosen is good. If the ground connection chosen is not good then testing while using that ground connection will give inaccurate results. The same is true of meters.

Similarly, when using the test light clipped to power (such as the solenoid's battery connection or battery +, the test light should first be checked by touching the tip to a known good ground to confirm that the light works.

This may seem simple but anyone who has done much testing will admit to having chased their tail looking for a problem which wasn't there because the ground or power connection chosen was not functioning.

Some terms worth considering:
1) Ground- is a connection, circuit or path which uses the vehicle's or unit's chassis. Modern vehicles have the ground connected to the battery's negative and are thus referred to as negative ground. Any connection making its way to the negative will be referred to as a ground. This is true of good or unintended connections. A fault in the wiring, such as a wire which has insulation rubbed through so that it is in unintended contact with the chassis will be said to have a ground (fault).

2) Open- is a point in circuit which does not complete as intended. In other words the circuit is open as a bridge can be said to be open. In either case there will be no flow of either current or traffic. A switch in the open position is off.

3) Short- is an unintended path which bypasses part of a circuit. This bypass will generally mean that the component being bypasses will not operate and because the resistance of the component is not present in the circuit, the current flow will be higher than intended. Typically short circuits or "shorts" will blow fuses or cause wiring fires. Shorts may be a short within the insulated (power or positive) side of the circuit, or the short may be an unintended path to ground which is properly referred to as a short to ground but is typically referred to as a ground.

Also, when the ground wire is removed from the bolt on the frame the light on the
tester still lights when I have it connected to the (-) negative on the battery and touch the
center of the lighter/adapter.

If the test light is clipped to a good ground, the test light has a path to ground which will allow the test light to light if the tip is placed in contact with a power source. In this case the power source is the centre (Canadian spelling) of the cigarette lighter adapter (socket) which means that power is present to the cigarette lighter/adapter because the test light obtains power from that circuit as indicated the light. The ground circuit of the cigarette lighter/adapter is not required to provide power to the centre connection of the cigarette lighter/adapter so whether it is connected or not will have no effect on whether the test light lights when connected to the centre connection of the socket. This is a normal result.

I suggest that you connect the test light's clip to the battery positive or the battery connection on the starter solenoid. Next, touch the light's tip to a ground such as a bolt on the frame or engine. If the light lights you know that the connection to the positive works and can proceed to test other grounds. Touch the tubular (as opposed to the centre) connection of the cigarette lighter/adapter and the test light should light (glow) to indicate that a path to ground is present through the tubular (ground) part of the lighter/adapter.

I tried two lighter chargers with my cell phone and got one to work sort of. It worked for a few seconds charging but then stopped. I don't understand why and if the ground wire is working.

In my experience the most likely cause of intermittent charging is that the cigarette lighter/adapters are typically of poor quality both the socket adapters and the accessories which plug into the socket. Usually testing by wiggling the wiring will indicate if there is an intermittent connection within the socket/adapter and if not the likely culprit is a poor fit between the adapter and plug. Looking and wiggling will usually narrow this down. Swearing is not optional during this aspect of testing and the more "blue" the air turns, the more quickly the problem can be located (VBG) or at least it seems that way.