Could I make the own USB Type A/A cable

The black connection provides power and data. The red one is power only. The second cord should be used if the black one doesn't provide enough power to power and/or charge the device. You'll see a lot of external hard drives come with the same type of plug (though usually both are black).

EDIT: To answer your second question, plug the device in using the black cable. If it runs normally and charges (assuming this has a battery), then you're fine. If, however, the device doesn't operate correctly or doesn't charge, then you will need to plug the red cord into another USB port. You wouldn't cause any harm to the device by plugging in both cables. Honestly, the red one would even be plugged into a different computer since it's just power only.

How is a USB-Y cable designed?

It is designed for devices which require more power than is available from a single USB port.

Whereas from the third test, it shows the male plug attached to thin cable doesn't carry enough power or usb data

This extra connector does not have any data lines (D-, D+); it only has the power lines (GND, VBUS).

Are all the three plugs meant to be connected to USB power and data?

Yes. The two connectors on the Y end should be connected to the PC/Laptop and the other end to your USB device.

I want to check if the cable is faulty.

First, I connected two male plug ends to laptop and third male plug to disk, it works fine.

• That is how the cable should be used so it is not faulty

• The other tests are using the cable in a way it is not meant to be used so the tests are meaningless.

What are the 2 different kinds of Y-Cable?

• One male connector, two female connectors

  There are two types of the one male two female cables. One is a basic charging cable, splitting one port's power across two devices. The other is a specialised cable that somehow splits the data lines - and it's only used in very specific applications.

• Two male connectors, one male or one female connector

  What happens is you have your USB cable, one side for the host and one for the device, but with an extra connector attached. This extra connector does not have any data lines (D-, D+); it only has the power lines (GND, VBUS). It's attached in parallel to the existing cable. In other words, VBUS is connected to VBUS and GND is connected to GND.

Source Can I safely connect the power-only-end of a USB-Y cable to an other power source? answer by Bob)

Power requirements

Some devices, such as high-speed external disk drives, require more than 500 mA of current[89] and therefore may have power issues if powered from just one USB 2.0 port: erratic function, failure to function, or overloading/damaging the port.

Such devices may come with an external power source or a Y-shaped cable that has two USB connectors (one for power and data, the other for power only) to plug into a computer. With such a cable, a device can draw power from two USB ports simultaneously.However, USB compliance specification states that "use of a 'Y' cable (a cable with two A-plugs) is prohibited on any USB peripheral", meaning that "if a USB peripheral requires more power than allowed by the USB specification to which it is designed, then it must be self-powered."

Source USB