Starting a sentence with "rather"

I've sometimes heard people use rather for connecting two sentences where the second one sets counterexample to something negated in the first.

This is not a meaningful sentence. Rather, it's an example.

First, I would like to find out whether this is a proper usage of the word. Finally, if it is, I would like to find out how to properly write such sentences (the above example is likely grammatically incorrect). I am especially interested whether they can, or even should for that matter, be separated by a semicolon rather than by a period.

Rather than focusing on grammaticality, which is not an issue, you should think of style.

The use of rather (with the meaning of “on the contrary”) as the first word of a sentence is well established. It is even featured in the example chosen by the New Oxford American Dictionary:

There is no shortage of basic skills in the workplace. Rather, the problem is poor management.

Hence, the main issue is that of writing style. Using rather as the first word in the sentence might tend to lead to shorter sentences; in this particular example, you could combine the two sentences into one. Depending on context, short sentences may make the writing clearer and closer to the spoken word, or seem excessively fragmented.