Looking at your video, (very good by the way) I think that the problem is that your Solar Panel 'controller' is probably doing what it has been designed to do. It probably has what is known as 'short circuit output protection. That means that when there is a short circuit across the output it will shut down to protect the solar panel, the equipment that it is supplying from further damage and itself.
A multimeter set up to read current (A) is effectively a short circuit, but when it is set up to read Voltage (V) it is a high resistance.
In the 'How to test Solar Panels ' video that you posted, there is no controller connected to the panel. They tested directly from the panel output itself. This is OK if you want to check the panel is capable of supplying maximum current, if you think that it is faulty but it is not how it works in real situations. You need to protect the panels and the equipment that they are connected to. That is why there are controllers in between the panel and the output.
Note: If your solar panel controller also has a regulated Voltage output (Voltage is never more than 12-13V DC) then the current supplied to the battery may depend on the voltage that the battery has.e.g if the solar output is 12.3V and the battery is 12V then the battery is only being charged by 0.3V and the charging current will be small.
First make sure that both solar panels are connected from +ve panel output lead to +ve controller input terminal and -ve panel output lead to -ve controller input terminal (as you probably have)
The way to test the output current that is charging your battery is as follows:
1. Measure the solar panel controller output Voltage - try to get maximum voltage by angling the panels. It may be that you can never get more than 12 -13V
2. Measure the battery voltage. - hopefully it is less than the solar panel controller output voltage.
3. If it is proceed.
4. Connect the -ve solar controller output lead to the -ve battery terminal.
5. Connect the +ve solar controller output lead to one lead of your multimeter. (the meter should be set up to read Amps/DC range 10A, the leads in common and 10A terminal - as you did correctly in the video)
6. Connect the other lead of your multimeter to the +ve terminal of the battery
Your meter is in series with one output lead from the solar controller i.e solar controller +ve output - meter in - meter out - battery +ve terminal.
7. If your reading is less than 0.2A (<0.2A) change your meter range to the Amp/DC 200mA range and change the lead from the 10A terminal to the other terminal on the meter to get a more accurate reading.
If you do not get a reading this way, (check the solar output voltage again to make sure it is more than the battery) perhaps then your multimeter may be faulty when reading current. Check the user guide of the meter in case there is a fuse (in the meter) on the 10A range. In most meters the 0-200mA ranges have a fuse in the meter (to protect it in case of higher current flow) but the 10A range does not. Maybe though yours has. Just a point to check.
Check to see if you have an adjustable voltage output option in the controller. If not and it is a fixed 12 - 13V output, the charging current will always be small as long as the battery that you are charging has a voltage close to the output of the controller. The charging voltage always has to be higher than the voltage of what is being charged otherwise the current will want to flow back the other way. Most controllers also have 'reverse voltage protection' which stops current flowing back from a battery into the controller if the battery voltage is higher than the controller output voltage. This may occur if a cloud obscures the sun for instance, and the solar voltage output falls.
Ideally to effectively charge a battery such as you have in the video, the output from the solar controller needs to be in the range 13.5 - 14V. Once the battery is charged, having what is known as a 'trickle' charge ( solar output just above battery voltage) will maintain battery at peak level, if it is not being used for long periods of time.
Hopefully this is of some help.