trim Ideal doesn't trim all the way

From Nick Addington via Google groups:

I ran this code:

R = QQ[x,y,X,Y,Z, Degrees=>{1,1,0,0,0}]
I = ideal(y*X-x*Z,x*Y-y*Z,x^2,y^2,x*y,Z-1,X*Y-Z^2,X-1,Y-1)
trim I

The output was

ideal(Z-1,Y-1,X-1,x-y,y^2,x*y,x^2)

when I would have expected

ideal(Z-1,Y-1,X-1,x-y,y^2)

Bizarrely, if I run

J = ideal(Z-1,Y-1,X-1,x-y,y^2,x*y,x^2)
trim J

then I do get the expected output.

Is this a bug, or should I tweak my input somehow?

I believe this is a bug, both in the graded and non-graded case:

i1 : R = QQ[x,y,X,Y,Z];

i2 : I = ideal(y*X-x*Z,x*Y-y*Z,x^2,y^2,x*y,Z-1,X*Y-Z^2,X-1,Y-1);

i3 : trim oo

                                         2        2
o3 = ideal (Z - 1, Y - 1, X - 1, x - y, y , x*y, x )

i4 : trim oo

                                         2
o4 = ideal (Z - 1, Y - 1, X - 1, x - y, y )

i5 : isHomogeneous  I

o5 = false

i6 : R = QQ[x,y,X,Y,Z, Degrees=>{1,1,0,0,0}];

i7 : I = ideal(y*X-x*Z,x*Y-y*Z,x^2,y^2,x*y,Z-1,X*Y-Z^2,X-1,Y-1);

i8 : trim oo

                                         2        2
o8 = ideal (Z - 1, Y - 1, X - 1, x - y, y , x*y, x )

i9 : trim oo

                                         2
o9 = ideal (Z - 1, Y - 1, X - 1, x - y, y )

i10 : isHomogeneous I

o10 = true

Reproduce the two trim examples from the issue in Macaulay2, including both the graded and non-graded rings. Then locate the implementation of trim and determine why the first invocation retains redundant generators. Done means both examples produce the reduced ideal on the first invocation, with regression coverage for the reported inputs.