People need to pay attention to the context.
It's a trust that provides a lot of mental health related services. "Lived experience" is a term already in VERY wide use in that sector to refer to the experiences and voices of people who know what it's like to be seriously mentally ill (and to get treated like shit a lot of the time by services), and more specifically, to describe roles in the service for who have those experiences and use them as part of their job working in the sector.
If an organisation advertises for a role using a particular phrase that might be ambiguous to the general public but means something specific to people who work in that type of organisation, then that's probably what they're using it to mean.
As the Trust provides other services outside mental health, perhaps the director will be expected to incorporate the perspective of people with lived experience of the other types of problem they deal with, too — it's not that clear (which isn't great, no — the ad could be better-written).