One useful definition of 'digital system' I like, is 'a system whose inputs, outputs, and internals have a finite number of valid discrete states'.
According to such definition, sound card output is digital, just because it has finite number of valid discrete states (for example 16777216 discrete amplitude states and 48000 states per second clock states)...
The output of a DAC is still digital, but add a capacitor or a
low-pass band-pass filter (often DC is blocked!), and you have an analog output, because it is continuous and no discrete states can be observed a priori in the output.
Let's take for example a simple logic gate AND. It has input and output which is continuous in time.
So, using such "crude approximation" it leads to analog category for digital logic gate...
No, it doesn't. While the input and output are voltage levels, in normal use one range of voltages is considered "low", another "high", and the rest are indeterminate. The
logic is discrete, while the voltage levels used to convey that logic is continuous.
This is why it is so important to consider the system at the correct complexity level. Any definition is useless if you only use its "holes" or weaknesses, instead of its strengths. Definitions are tools, not some kind of absolute revelation of fundamental facts.
As tggzz explained, many logic gates can be used as an analogue component, too. Thus, "discrete" or "continuous" also depends on the design, the intent, of the component and system.
And another example with analog switch connected to some clock to enable its output at discrete points of time.
Since it's discrete in time, using such "crude approximation" it leads to digital category for signal which is pure analog...
No, like I wrote: there are mixed systems, but we lack exact terminology for them. I thought I made this clear with the triac mains dimmer with a switch: the states consist of both discrete (off) and a continuous range of states (duty cycle). The clocked part is digital, because it has a finite number of states; but because of its construction, it can also be used with analog signals. Thus, a mixed system.
The reason I prefer this definition over others is that it is
useful. Using that definition, it is very easy to branch into mathematics (discrete side to mathematical logic and various theories, continuous side to functions, calculus, transforms), physics (discrete vs. continuous systems, mechanics), electronics (especially the theoretical side), and even history of computing/calculation and esoteric systems like fluidics. You can find "grasping points" in each using the terms in the definition.
Sure, if you stop nitpicking and use the definitions as I described, you can poke valid holes into it. That does not make this definition any less valid in the above sense,
because it is useful. We haven't discarded Newton's Laws of Motion just because we discovered that they're not exactly correct in all situations; they're still a valid model within their known limitations.