I bought a 2003 Type S last year, and have been trying to figure out the cause of the transmission problem for quite a while. I've replaced all of the external solenoids, filter, solenoids, and replaced the ATF with Honda juice.

Here's the problem:
1. Occasionally when I approach a stop, the transmission "forgets" to shift to 1st. I can move the shifter so that the transmission finds 1st gear.
2. At full throttle in 2nd gear (maybe it's 3rd), instead of downshifting the transmission behaves as if it is in neutral (rpms increase, but no acceleration). The problem doesn't occur in higher gears at full throttle - downshifts are normal.
3. Using the manual shifter to upshift or downshift, there is no slippage in any gear.

The last bullet suggests to me that the problem is external, not internal. Could a faulty Transmission Position Sensor/Switch cause this problem?

 

for point 1, is it behaving similar to that shown in this video?

 

Yes! He posted a link in the comments, but it no longer goes to the Honda bulletin. He does mention that the problem is caused by an internal clutch. That sucks.

 

I did find recall bulletin 04-20, automatic transmission 2nd gear inspection. The issue is heat buildup between the countershaft and secondary shaft 2nd gears. The solution is to add an oil jet kit that supplies oil through the atf filler port.

The bulletin says gear tooth chipping or breakage can occur, but doesn't mention the downshift problem as a symptom.

My car does appear to have a remanufactured transmission - I don't know when or who installed it or when. It doesn't have the jet kit mentioned in the recall.