How to delete, including the current character?

No. Backward motions always start on the left of the current character for c, y and d which is somehow logical but also unnerving.

The only "clean" solutions I could think of either imply moving to the char after c first and then do a forward delete:

Tcde

or using visual mode:

vTcd
v3hd

But, given your sample and assuming you are entering normal mode just for that correction, the whole thing sounds extremely wasteful to me.

What about staying in insert mode and simply doing ←←←←?

try this:

TcD

this will leave abc for your example... well if the abcdefg is the last word of the line.

if it is not the last word in that line, you may do:

ldTc

or golfing, do it within 3 key-stroke:

3Xx or l4X