I beleive your rear wiper circuit is controlled by two relays. It sounds like you bypassed one relay when you tapped into the amp wiring. Also, current through the black/light blue wire is controlled by a relay. It would not be continuous. It would "pulse" as you call it. Both the 3-wire motor circuit and the 4-wire motor circuit use 15 amp fuses. Probably not a good idea to be using a wire on a 25 amp fuse for this circuit.
I think I understand how the 3-wire motor works now. The 3-wire motor circuit uses only one relay. The relay allows a current "pulse" through the violet/yellow wire into the motor and out through the black wire to ground. This causes an internal switch to move from the "park" position to the "run" position. This allows a continuous current on the brown/white wire to flow into the motor and through the black wire to ground (actually it's "pulsed" internally because of the internal switch). The wiper goes up. When the relay stops the current pulse through the violet/yellow wire, the motor's internal switch moves back to the "park" position, the current from the brown/white wire stops, and the wiper goes down to the same position as being parked, because on this window design, it has no need to lower itself off the glass.
If go back to your original wiring configuration, check your fuses to make sure they are all the right amps and in the right place, and your relays are functional, I think this might work. I am assuming your 98 Explorer has the same circuit components in the same positions as my 99. You should verify everything yourself.
1. Disconnect the negative battery terminal and remove the rear wiper down relay from the battery junction box under the hood (relay slot #8). The rear wiper up relay (relay slot #10) will act as the single relay from the 2002-2003 rear wiper circuit.
2. Connect the violet/yellow wire of the 3-wire motor to the white/orange wire that previously went to the 4-wire motor. The violet/yellow wire that previously went to the 4-wire motor will not be used.
3. Connect the black wire of the 3-wire motor to the black wire that previously went to the 4-wire motor.
4. Connect the brown/white wire of the 3-wire motor to the black/light blue wire that previously went to the 4-wire motor. With the wiper-down relay removed, this wire will no longer get "pulsed" current. We will set it up to get continuous current.
5. The empty rear wiper down relay slot has five terminals - two horizontal and three vertical. The two horizontal terminals are #3 (the other end of the black/light blue wire) and #5 (a series of white/violet wires that connect to the 15 amp fuse in mini-fuse slot #8). Connect a jumper wire or cotter pin between terminals #3 and #5.
6. Reconnect the negative battery terminal, turn the key, and turn on the rear wiper.
If it works, you should make arrangements at some point to replace the jumper wire or pin with a more solid connection between the black/light blue wire and the white/violet wire inside the battery junction box.
If it does not work, then I have thoroughly wasted your time and mine. On paper, it looks like it would work.