What exactly is a signal in Engineering? ... Does the definition mean that anything that can change in a circuit is a signal?
I don't have a dictionary definition but I think you can consider a signal as any modulation of an energy source that conveys information rather than power. This covers such cases as transmission of information, digital or analogue, by electrical, optical, audio, fluid pressure, mechanical coupling, etc.
... in a switch circuit, the input signals are the states of the component switches ...
I'd say the signals are the voltages provided by the switches.
... and the output signal is the state of the whole circuit itself ...
Again, it's the voltages output by the whole circuit, but you've got the right idea.
... while in a logic circuit, the input signals are the voltage at points that are not behind any gates(and also behind a switch? I'm still not sure about this) and the output signal is the voltage at the point behind all logic gates, with respect to the Ground.
Don't get so hung up on this. The input signals have to come from somewhere and may be as a result of some previous logic, simple switches or even (in contradiction to my "definition") from a permanently high or low logic level.
A signal doesn't necessarily have to be referenced to ground. Some signals are differential such as RS485 and USB where a pair of wires are used and swap polarity with each other to provided the signal.
In a circuit that contains both switches and logic gates (not like just a few switches followed immediately by logic gates but maybe a bunch of switches connected in series and parallel and then is connected with a combinational logic circuit), what exactly are the input signals and output signal?
Electrically there is no difference so it really just depends where you define the boundary of your system.
I mean, is it possible for the input signals and output signals to be of different types?
Sure. For example, a logic-level signal could be converted to a higher voltage or to an optical signal for fibre-optic transmission.
From the comments:
... but when 2 switches are connected in series if the first switch is open then no matter what state the second switch is, the voltage between it and the GND is always 0.
simulate this circuit – Schematic created using CircuitLab
Figure 1. Switch logic feeding transistor logic.
For a mechanical switch circuit to feed a transistor logic circuit it is important that the logic input is not left floating where it might assume a logic 1 or 0 or indeterminate value. For this reason the logic would require a pull-down resistor so that if one of the switches is open that a definite 0 is received on the buffer input. When SW1 and SW2 are both closed a logic 1 is forced on the input despite the pull-down resistor. (That's the error in your reasoning. CMOS input impedance is so high that it doesn't pull the input to zero. The input voltage is floating and could be any value.)
That's probably why I think the input signals of a switch circuit are the state of the switches rather than something "voltage-related".
The switch status results in a voltage. The transistor logic only knows about the signal voltage and knows nothing about the switches.