A p-value, which measures how surprising the observed data would be if there were truly no effect, of greater than 0.05 does not, they say, show there is 'no effect'. … Instead, the authors say, a non-significant result simply means there is insufficient evidence to conclude a difference exists, not proof that a difference does not exist.

That's a pretty bad mistake. Taking "not proven" as meaning "proven not". It's like they either never learnt logic or they don't understand their statistics