many people don't take
enough "sap wood" with the bud when readying a bud graft.
What you want is all the bud and some of the underlying heart wood when you are taking the bud to graft.
From there you use the T slit, remove that heart wood, leaving all the cambium, (I like to also peal the outer bark just before I put the graft together, this gives fresh material to touch fresh material and that helps the bud grafts to take) and slip that into the T slit.
From that point it is just wrap it up.