The starter circuit relay receives a ground to start the engine, either from the neutral switch, or from the clutch safety switch (with the clutch lever pulled).
Sounds like you're denied the latter; i.e., your clutch safety switch is inoperative. You can work around this problem by defeating the switch, with a wire segment or mini-fuse, as shown on the website threads, or fix/replace your clutch safety switch.
Irrelevant anecdote follows: I recently had the OPPOSITE problem; I could start the engine with the clutch lever fully pulled, but NOT with the transmission in neutral. Incidentally, no green neutral light, either. Cause? Using an electric impact wrench to torque the countershaft sprocket nut, the severe vibrations yanked the neutral switch wire connector from the neutral switch. Thus, the only ground possible for the starter circuit relay had to come from the clutch safety switch, until the neutral switch wire was re-connected to the switch terminal.