Invisible zippers come in 3 weights: #2, #4 & #6
#2 are the standard that you find in fabric stores. (2mm across the coil)
#4 or 4mm across the closed coil, are a little heavier & stronger. Use in bride's maid dresses, pillows.
#6 or 6mm across the coil, are the size that Vera Wang uses in her gowns & I use in cushions or slipcovers.
To make your selection of the correct size invisible for your project a little harder, some manufacturers call these #3 & #5 invisible, instead of #4 & #6. My coil is a bit heavier but will normally fit #3 and #5 invisible pulls.
The little plastic invisible foot that you have will NOT work with the #4. Being cheap, I just use my standard zipper foot & stitch in the ditch. If you have a commercial machine, I stock Sam Sloan's adjustable foot in "Thread & Workroom Supplies".