I believe the thistle you are referring to is actually the Common Sow Thistle (Sonchus Oleraceus).
The sap is white but quickly turns brown on exposure to the air. I too have used it for many years to treat minor skin cancers and sun spots and know it to be safe and effective.
I have fair skin and spent much of my youth on the beach in summer at a time when there was little or no protection from the sun. By age 30 I was regularly going to the doctor to have BCCs burnt off with liquid nitrogen. It was a effective but painful and always left a patch of de-pigmented skin. Then one day I met an old horse trainer who showed me how he trated sun cancers on the nose of his horse by breaking the stem of a common weed and dabbing on the white juice that oozed out. The weed is known here as Milk Thistle or sometimes St Mary's Thistle and is prolific in this part of the world; it pops up everywhere, even in cracks in the pavement.
I never went to the doctor again for BCCs and I have treated dozens of them over the last 30 years or so using the Milk Thistle. You simply break the stem near the top where it is soft, wait a few seconds for the white sap to rise, and dab it on. Then leave it. After an hour or so it will turn a brown colour and after a few days will turn almost black. It sticks like tar and will not come off, even in the shower, providing you dont scrub it! After a week or so it will gradually wear off and the BCC will be much reduced if not gone altogether. Persistent cases may need a second application. It is painless, does not leave a scar or loss of pigment; it is simple to use and it's free! The whole world should know about the old fashioned "cow-cocky" remedy, but I guess that would put a lot of fancy skin cancer clinics out of business.