Wolfram Language (Mathematica), 212 170 166 164 159 bytes
A_~J~B_=Fold[Switch[{a,b}=#;#2,"f",{A,b},"F",{a,B},"e",{0,b},"E",{a,0},"p",#+Min[a,B-b]z,"P",#-Min[b,A-a]z,"o",Echo@a;#,_,Echo@b;#]&,z={-1,1};0z,Characters@#]&
Calling J[A,B] returns an interpreter for the (A,B)-bucket system, which can be applied to a program with J[A,B]["prog"]. The "o" and "O" commands print using Echo, and the return value of the program is the final bucket state.
The program is a big Switch statement folded over the characters of the program, step-wise updating the contents of the buckets. Dispatching the Switch takes a shortcut by defaulting to "O", as the spec says that there are no characters to be expected other than "fFeEpPoO".
Debugging is done by replacing Fold with FoldList, so that the return value of the program becomes the sequence of bucket states traversed during program execution. Un-golfed version:
J[A_,B_][prog_] :=
FoldList[ (* debug version *)
Switch[{a,b}=#; (* initialize a and b to the current bucket contents *)
#2, (* switch as a function of the next character in the program *)
"f", {A,b}, (* f: fill first bucket, leave second bucket unchanged *)
"F", {a,B}, (* F: fill second bucket, leave first bucket unchanged *)
"e", {0,b}, (* e: empty first bucket, leave second bucket unchanged *)
"E", {a,0}, (* E: empty second bucket, leave first bucket unchanged *)
"p", {a,b}+{-1,1}*Min[a,B-b], (* p: pour first bucket into second bucket *)
"P", {a,b}+{1,-1}*Min[b,A-a], (* P: pour second bucket into first bucket *)
"o", Echo@a;{a,b}, (* o: print contents of first bucket *)
"O", Echo@b;{a,b}]&, (* O: print contents of second bucket *)
{0,0}, (* start with both buckets empty *)
Characters[prog]] (* fold over the characters in the program string *)