A breakdown in communications with the Tower contributed to a near runway incursion after landing by an air carrier flight crew in STL.

Narrative

STL was reporting approximately 4;000-5;000 overcast with about 4-5 miles of visibility in light to moderate rain. It had been raining for several hours. The First Officer wisely selected medium brakes and flew a perfect approach and touchdown. When the First Officer applied manual braking at 100 KTS I noticed the runway was slightly slippery due to the rain. I took control of the airplane at 60 KTS just before a high-speed turn off. Under dry conditions; making the high-speed turnoff would have been no problem. I elected to take the next one that was a 90 degree turn. As I took control of the aircraft ATC told us to exit at the high-speed. I told the First Officer I will take the next one and he told ATC 'we'll take the 90; cleared to cross Runway 12R'.The ATC call came at a time when I was more concerned with getting the aircraft safely off the runway than hearing everything the Controller said and I was more concerned with ATC knowing that we would NOT be taking the turnoff he instructed us to take. The Controller came back by stating that there was traffic on a 1 mile final behind us. I clearly heard the First Officer state that we were cleared to cross 12R to the Controller who did not challenge it. The 90 degree turnoff makes for a quick arrival at 12R and as the nose of the aircraft approached 12R the Tower Controller stated that we were cleared to cross 12R and that our instructions had been to contact Ground Control. I assume that means that Tower never had cleared us across 12R BEFORE that time; but the Controller never challenged the First Officer on his readback.A different air carrier Captain was sitting in the jumpseat at the time and we had the speaker on to keep him in the loop. We all discussed it after arriving at the gate and agreed that we could not remember the exact wording of the Tower Controller and as to whether he had cleared us across 12R prior to our approaching it. We were all sure that the First Officer had stated that we were cleared to cross and the Controller never questioned his readback. Technically the Tower Controller gave us our crossing clearance just prior to the aircraft crossing the runway and we had already cleared the runway visually both directions.ATC is notorious for talking to us at that critical moment on the runway when the Captain takes control of the aircraft and we are trying to taxi clear! Very difficult with a normal day/VFR day and makes it even harder with lower weather conditions. I should have been more alert to the clearance we were given. We were at the end of a long 4-day trip and ready to go home. I also take responsibility for NOT questioning the clearance prior to crossing the runway when all I had heard was the First Officer's readback and the response to the readback.