Is it possible to make a new width profile?
I want a stoke that will only "backfill" into the object, the one below is the closest I can get but the stroke still protrudes out the shape slightly and also has the tapered edges.
Any help much appreciated.
Yes, use the Width tool the create the profile you want and then use the little save icon in the width profile drop-down.
"stroke still protrudes out the shape slightly and also has the tapered edges. "
If you are asking how to create a new width profile, then Doug gave you the answer. If you are asking how to make it appear the way you want, you might double-click each width point and type the numbers you want. They don't have to be the same on both sides. Please clarify.
See these links for more info:
Width profiles are made with the width tool. They can be saved at the bottom of the Stroke panel. If you don't see that area in the stroke panel, please go to the panel menu, in the upper right corner of the panel (3 or 4 little lines) and select "show options"
About what application are you talking here?
I'm just asking because in the desktop version of Illustrator I've never seen a panel like the one you show in your initial post.
So it is! I must be needing glasses.