There are many web puzzle to identify, but does it really have any scientific proof?
something like - http://braintest.sommer-sommer.com/en/
any logic behind it?
In general I'd say "left and right brained behaviour" is total nonsense. I guess the origin of this kind of claim is a bad press release or something.
Let's start with something simple. Say hearing. When you hear a sound to your left ear, it is projected strongly to the contra-lateral side (right hemisphere), and weakly to the ipsi-lateral (left hemisphere) of your brain. So the brain response to a sound presented to the left ear is lateralized. Typically, this means that the other hemisphere is "more" responsible for processing the input. Similar thing happens for the other senses too.
So what is interesting about lateralized brain responses? Well, it would be very interesting to know for example are memories lateralized. It would tell something about the system: is it a kind of distributed system, where parts of a memory are scattered here and there, according to some logic, or is a single memory stored on a small area (and maybe only on the other side of the brain), etc. The distributed hypothesis seems more plausible according to recent research.
Let's talk about handedness next. Are you right handed? Or left handed? Or ... something between? Actually, even something like handedness is not a binary thing. It is more like a value on a range from 0 ("very left handed") to 100 ("very right handed"). This is taken into account in decent research, where ~1 hr per participant is used to assess that value (if it is relevant to the study).
So we have now talked about lateralization. How about the "left and right brained behaviour" then? In the URL you have provided it is implied that certain behavioural attributes are somehow related to lateralization. Lateralization of what? All the attributes, e.g. "being rational" depend on so many things: seeing, hearing, thinking, retrieving memories, ... that saying such a thing is not plausible.
Anyway, many brain processes are lateralized. That is the word to look up for for more information. But a thing like "behaviour" is too complex to be put to one hemisphere.
Reference: Accumulated wisdom from being a research assistant in a neuroscience project. (so you should trust pubmed.org more than me).