When making your sheet bend, if instead of passing the rope end through, you pass a bite of the rope end through (for ease of undoing) does this make the knot a double sheet bend?
No, very definitely not. A double sheet bend involves taking a second round turn round the two parts of the thicker rope. What you describe is a slippery version of the single sheet bend.
The fisherman's knot - a thumb knot tied in each end around the standing part of the other end - is very secure, especially in its double version and because it grips the rope rather than relying on a kink it gives a much higher percentage of the strength of the rope. BUT - it can be very difficult to undo after taking a heavy load.
A couple of years ago I broke some ropes trying to tow a bogged-down lorry. I had used double sheet bends to join equal diameter ropes and the main cause of failure was the sheath parting and allowing the core to slip through. The rope that failed was the one that would normally be the thicker one, i.e. the one formed into a simple bight. BTW the rope in question was 14mm Dyneema so it was a really serious load!
When a rope has a sharp bend, then the fibres on the outside of the bend are in tension and those on the inside tend to be in compression. Put that rope under load, and the tension in the outer fibres increases. Stretchy ropes tend to even out the loads. Dyneema, with little stretch, can't even out the tension and hence almost all the load is carried on the outer fibres. A knot in nylon may lose 40% of the strength of the rope; the same knot in Dyneema (or Kevlar) may lose 70% of the strength.