Pindar took over the role at the BPIF in July, but steps down with immediate effect.
Middleton said that he would be resuming the role until a new president could be found, although no timeline has been agreed yet.
He said: "It is very sad to see Andrew go. We want to ensure stability during the handover. Kathy Woodward is doing a great job and by stepping back in it allows her to continue what she is doing while the BPIF gets on with the next stage of its development."
Woodward said: "Andrew and three generations of the Pindar family have freely given their time and expertise to the BPIF and we would like to thank Andrew for the significant contribution that he has made over the years, particularly his involvement in developing Vision in Print.
"I look forward to continuing to work with Rupert and the BPIF Board to implement the forward strategy of the BPIF which was signed off by the National Council in May."