James ,
You are right that not all carpets where you can see the pattern in the backing are BW's and most can be cleaned without problems.
Another way of determining wool or wool mix from Polypropylene is to wet a little with your cleaning solution .
PP will not wet out very well whereas wool will soak up water.
Another clue is the feel , PP is harsh and will cause a lot of friction , wool is much more gentle.
As John says the burn test will produce differing results but becomes less clear on mixes as the plastic part will melt.
Lots of potential problems can be spottted by checking how well fitted a carpet is , has it come away particularly near rads, doors etc.I have seen BW's which have shrunk due to heat , general relaxation and have never seen water!
Always be attentive if you are not sure , a carpet which is going to shrink will start to go within a few minutes.Don't start in the middle , find an inconspicous place , clean a little watch , wait.
Cheers
Doug