Hello ft, glad to be a part of the discussion!
You are quite right:
T*sin (a) - D*sin (a) + L*cos (a) - W = 0
That's the total equation.
And my intention was of course to refer to (a) as 'climb angle' throughout the post.
But still my point stands. If one climbs with the same speed and the same weight as a similar plane would for level flight the angle of attack would be the same in both situations. Therefore I haven't included it in the equation.
Assuming then the airspeed is the same the force perpendicular to the thrust is the same. So the lift vector is the same.
The vertical component of lift - L*cos (a) - will on the other hand be less. By increasing thrust and hence the vertical component of thrust - T*sin (a) - one compensates for the reduced vertical component of lift.