I love my '02 Protege ES, which incidentally will have the same engine as the '01 LX that you are contemplating. Which tranny will it be - 5 speed or auto?
What I like about my car:
* fun to drive, good handling
* decent acceleration for its class
* generally good fit and finish, inside and out
* no reliability issues yet, no major warranty claims (only replacement of bolts that like to disappear from the manifold heat-shield, and a check of the A/C system), granted it's only got 36k on the clicker - brakes also wearing well, with over 60% left on the front pads.
What could be better, but not major issues:
* not fond of the white-faced gauges on the ES trim level, but that of course is personal preference
* would have liked a 4-spoke steering wheel instead of 3
* low-speed (under 20 km/h) roughness of drivetrain - a common issue with the 2.0L in either tranny
* lack of power lock switch on the passenger side
* lack of variable intermittent wipers
Gripes:
* not the greatest fuel-economy in it's class, but if I do a road trip, I can manage up to 32 MPG (using American, not Imperial gallons)
* A/C is weak, hence my taking it in for a check. Found to be within spec. Also a common gripe on the vehicle from other owners.
* horrible OEM tires - 15" Potenza RE92's seem to be the worst tires ever made for wet weather. The 16" Dunlops apparanently don't get too warm a reception from owners with those, either.
Overall, you can't go wrong with the FS-DE 2.0L engine Proteges - a good bang for your buck. Of course, Corollas are solidly built cars as well.
See which one floats your boat and go with it.