2, amplifies a signal ( does this mean voltage?)
Both, depending on the circuit. A transistor at it's heart is a current amplifier. In the linear range the collector-emitter current is proportional to the base-emitter current.
But, most amplifier circuits are voltage amplifiers... Let's say you have an audio amplifier with a gain of 10... That's the voltage gain. If you put 1 Volt in, you get 10 Volts out. But in fact, it's also amplifying the current... If you have a speaker connected you are getting more current out than you are putting in. But with nothing connected, you are still getting the 10 Volts out, but no current, so no current gain. Since the output voltage is always proportional to the input voltage (when everything is in the proper range and operating normally), and the current depends on the load, we classify it as a voltage amplifier.
If you open-up an electronic device, most of the time there is one power supply (or battery) powering all of the transistors and everything else.
I also learned that in a transistor the 'base' is what triggers the conductivity between the collector and emitter and that the 'C' and 'E' are interchangeable ? In terms of connecting them to a power source's pos and neg. ?
No, they are not interchangeable. The base-emitter voltage & current "control" the collector-emitter current.
Now the main question is, does the power source at the base need to come from a separate circuit that the power that is at the collector and emitter?
No. In fact you need a common connection, usually a common ground. If you want to boost the Arduino output to 12V, you might use a separate 12V supply or you might use the same 12V supply that's powering your Arduino. But, "separate" does not mean "isolated" because you need that common ground. (You can use an optical isolator if you need isolation.)
If this is true, What if I have a 12v power source that I am using to power the entire circuit and arduino and relays etc ?
Usually that's fine. But with relays & motors, sometimes you can get electrical noise on the power supply that will cause a glitch or crash your Arduino. Just something to be aware of.. There are "tricks" with diodes & capacitors that can help to block/isolate the noise.
Lastly, in the attached images, the author is using the same type is transistors, yes in one he says that a low triggers the transistor, whereas on the other diagram, a pos signals triggers it. Is there a mistake is any of the schematics or am I mistaken with something ?
The author is made a mistake! Both circuits are the same! It can get complicated, and you have to analyze the circuit in by figuring-out the base-emitter current/voltage and then what happens when the current gain is applied through the collector-emitter. In circuits where the emitter is not grounded, the emitter voltage changes depending on the emitter current, and the voltage applied directly to the base may not be equal to the base-emitter voltage.
And, because of the inverse relationship between the collector-emitter current and the collector-emitter voltage, most amplifier circuits are inverting amplifiers (i.e. +1V into an inverting amplifier stage with a gain of 10 will put-out -10V). In the switching examples you've shown, the voltage logic is inverted, and it's the collector being "pulled down" nearly to ground that creates a voltage drop across the lamp/relay and turns it on.