positive thinking and illness

This is a tricky area. There is plenty of solid research that does support the idea that positivity, being optimistic, or a strong mental attitude does have some effect on physical health outcomes. This isn't magic; it is mind-body connection. The perfect example of the mind-body connection that most can understand and accept is how stress harms us physically... and can even kill us. So let's accept that there is some effect here.

People who follow this way of reasoning, always prove themselves to be 'right': if there is no cancer recurrence, the person was a positive thinker

We see this with religious belief all the time as well. If they are healed, it is the god they prayed to. The biggest fallacy here is Survivorship Fallacy. All the cancer patients who die aren't around to share their stories of how positive thinking and gods didn't save them, so we (the living) hear from the winners only.

I think you would enjoy (and find helpful) my article I wrote on positive thinking here: https://www.hostingauthors.com/posts/bobennett/bright_and_shiny__rethinking_positive_thinking.html

Sounds a bit like one of cum hoc ergo propter hoc or post hoc ergo propter hoc. The chances being higher lower is just that, it helps but isn't definitive, so claiming that the person wasn't positive is victim blaming. 
 

Perhaps a bit of Cherry Picking where you select the results which match with your views?

This would be known as confirmation bias.

Check out Daniel Kahnman’s book, Thinking Fast and Slow.