Further points:
The curve that you're trying to achieve is far from exponential. An exponential curve has a gentle curve all the way along, the curve you're targeting has a pronounced, not-at-all-exponential knee at the end of its fall.
Also, in trying to use the exp source, you fed in 350ms as your time constant (i.e., 100% - 36.8% time), not the time to fall to 10% as specified in the doc. Divide by -ln(0.1) to specify a fall to 10%; i.e., the time constant should be 350ms / 2.303 = 152ms. Also, t2 is supposed to be the time to transition into the fall; not 0. That will make the shape meet the spec, but it won't look the same as the picture, because again, the picture isn't exponential.
All of this aside, why are you so interested in matching the curve on that page so closely? That curve you're trying to imitate so closely isn't an engineering spec, apart from the specified properties: peak value, pulse width (10% -- 10%), and rise time. So reading so much into the exact shape of the curve is probably a bit silly.