A "case in point" is an example--it is a case (instance, occurrence, example) in defense of your argument--the "point" you are trying to make.
Edit: Without the actual context, I misinterpreted your sentences. You are citing "indenting at the same level" as an example of the highly popular beginner style--right? I would say the "classic example" or the "most common example": "The classic example here is the practice of..."
The "case in point" expression wouldn't typically be used in this structure. Here's how I'd phrase it if I were asked to use "case in point"--
"As a case in point, consider how frequently beginners indent continuation lines at the same level..."
I think you are mistaken in your belief that starting a sentence with
Being past participle
in isolation has an implication of
While he was being ...
In all cases I can think of we would explicitly include the extra information
While he was being taken to the hospital he died
While he was being beaten by his mother his father arrived
When we start the sentence with Being we are using the adjectival form
Being a doctor he was able to give medical assistance
Being a lawyer he was familiar with the legal process
Your example of
Being beaten by the snow, he died
is actually an example of the adjectival form of being. It is important to realise that beaten also may be a verb or an adjective. In this example we are using beaten as an adjective meaning defeated.
having been defeated by the snow, he died.
is the interpretation of
being beaten by the snow, he died.
You are thinking of this as if it were like your previous "mother" example, as if the snow were beating, that is hitting, a person.
while he was being beaten by the snow, he died
which is not the correct interpretation.
Best Answer
The more formal version is the following:
But at some point, this got shortened:
The short form now has informal and idiomatic usage.
Most likely, the confusion over adding is back in again occurs when people get stuck in an intermediate state between the original version and the shortened form.
So, we end up with this kind of situation:
It may be used by some people, but it's not really well formed. At best it's redundant, and at worst it could be considered asyntactic. (I would say ungrammatical, but that becomes a grey area when it's actually in use and understood; prescriptivists would call it ungrammatical.)
So, while 3. might be considered idiomatic in certain contexts, it's probably better to use either 1. or 2. And it's certainly better to stick to 1. in strictly formal writing.