A TRACON Controller conducting OJT reported an air carrier reported a NMAC with a VFR aircraft.

Narrative

I was training a FPL (Full Performance Level) on a temporary detail in this facility. They had been certified in this facility previously. Aircraft X was descending to 6000 ft. on a downwind vector for Runway X; and Aircraft Y was southbound VFR; assigned 4500 ft. I suggested to my trainee she might want to descend Aircraft X to 4000 ft. and coordinate it; in order to get Aircraft X below the VFR traffic; so that they could work the aircraft without having to worry about the VFR traffic being in the way. They descended Aircraft X to 4000 ft.; but also turned Aircraft X on a right turn vector. This was not something I would have done. My plan would have turned Aircraft X left. Aircraft Y reported Aircraft X in sight and was instructed to maintain visual separation. Aircraft X descended at such a rate that the two aircraft were separated by maybe 300 to 400 ft. when they converged. However; Aircraft X responded to an RA; while Aircraft Y advised he was climbing to miss the Aircraft X. I informed my supervisor that Aircraft X had responded to an RA. I was informed by my supervisor that the Aircraft X pilot had reported the event as a near mid-air collision after he had landed. In the situation; I pictured Aircraft X being assigned maybe a 10 degree left turn; and descended to 4000 ft; to get below the traffic without any converging course issues. I am sure that if I had been working the sector myself and not training someone; I would have handled the situation in such a way that there would not have been an RA nor a report of a NMAC. I recognize that because I was training someone who is an experienced CPC; and someone who I have worked with before in this facility; and whom I generally trust as a controller; I did not intervene in the situation when I should have. Hearing my trainee's perspective on the situation; it is apparent that she thought I had intended for her to take the action she took. I would have turned Aircraft X left; and she turned Aircraft X right. So; I recognize there was a misunderstanding and a miscommunication between my trainee and me; along with an expectation bias on my part.I think I can consider this a training situation where I allowed the trainee to go 'too far;' in taking an action I myself was uncomfortable with at the time; but where I was incorrectly relying on the experience; trust; and respect I have for my trainee; but wherein my trainee herself misunderstood my suggestion for her to take a specific action; and expected that what she did was what I was asking her to do. It was a combination of two different expectations and assumptions by the two of us; which were not well aligned nor communicated to each other. I take full responsibility for the situation; since obviously I was the trainer and I should have taken action to fix the problem.