Why are binary signals drawn continuous in time despite being a digital signal The conventional definiton of a Digital signal is as follows:
A digital signal is a signal that is discrete in time and quantized in amplitude. Almost all resources (textbooks, online sources,etc) stick to this defintion and gives emphasis to the point that a digital signal is discrete in time.

A binary signal is a  Digital signal.



But according to the defintion, the signal should be discrete in time to be a digital signal. In the plot, a binary signal is drawn as continuous in time. Then how can this be a digital signal according to the defintion. 

A digital signal according to the definition should look like this:


Also I read that a digital signal has different meaning/definition in different contexts.
For example the above mention definition is apt in the signal processing context, whereas in digital electronics where the binary signal mention above is used, a digital signal is a signal that takes only discrete amplitude values(ie. it is quantized in amplitude) and it can be continuous or discrete in time.

I am confused that why is the above definiton given a strict emphasis even in digital electronics context usually. Almost everywhere I look in internet and even some teachers give the conventional definition of a digital signal without mentioning the context.

Are my above findings correct?
I would like to know the exact definitions of a digital signal according to different contexts.
You are right and wrong!

If you would like to treat a binary signal as a discrete-time sequence having values of $0$s and $1$s, then you would plot them as all other discrete-time sequences, using the lollipop display.

However, for those electronic circuits that implement binary logic operators using physical devices (gates etc.), the associated voltage and current waveforms are analog in nature. What's binary there is the information that the voltage or current waveform carries. But the physical existance is analog. So if you want to plot them, then you would do it just like a logic analyser (oscilloscope) would draw them on their phosphorus crt screen...