I used to use the umbrella/specific definitions, even though they didn't make much sense to me. I ended up redefining them for my classes & friends.
I use transgender to refer to "someone who's gender identity does not match the one they were assigned at birth". Gender in this sense is the categorizing of people based on social norms and their sense of self. It includes expression, types of activities, and sense of ones self in terms of being male, female, both, neither or other.
Transsexual to me is "Someone who's sexual identity does not match the one they were assigned at birth". This means their body isn't optimal for their survival due to not producing the needed hormones, producing features that are disconcerting, etc.
I identify as both transgender and transsexual. I'm transsexual because although my body was assigned to the female sex at birth I needed testosterone to go through puberty. I'm transgendendered because although I was socialized to be a woman, I'm a man.
The way I see it transgender pertains to gender, or the mind & society while transsexual pertains to sex, or the physical traits.