Cu Chulainn
you better get the flight review signed in log and then 'company rental checkout' training flights signed and THEN nothing else mentioned (you put it as rental or something), else you're 100% gonna run into troubles with any EASA CAA claiming you were PIC because you paid rental rate and not training rate and company instructor was with you as 'safety pilot' collecting log hours.
To think otherwise, you'd be too naive. THe responsibility for safe conduct of flight and any emergency decision making, that'd NOT be you on flight with FAA CFI rated and legal company instructor. Google any interesting NTSB or FAA reports regarding any responsibility in incidents/accidents. One passenger CFI rated pilot did last minute save to prevent serious damage and 'manipulating controls' self-preservation bacon saving got him into trouble. Or the CFI rated friend of airplane owner in trouble for fuel starvation flight where he wasn't even legal to fly that time.
I trained and instructed in the US in the past and have been involved with EASA stuff, have kept track of various things, issues, considerations, legal stuff or practicalities.
In the eyes of FAA or IAA, UK CAA or whichever, instructor onboard that has dual controls installed (that'd be litmus test if whether the CFI is there for tips and ideas and solely for a nice view ride), is the acting PIC. End of story.