ACR TURNS WRONG DIRECTION TO EXIT RWY.

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ON LNDG AT DTW; WX WAS 200 OVCST; 1/2 FOG. LIGHT RAIN RVR 2400 (THIS WAS RPTED). LNDG CLRNC WAS LATE DUE TO PRECEDING ACFT EXITING RWY. TWR COULD NOT SEE ACFT (TWR IN CLOUDS) AND ASKED ALL ACFT TO CALL CLR. WE WERE GIVEN CLRNC TO LAND AT APPROX 500 FT AND HEARD PREVIOUS ACFT CALLING CLR ON FOXTROT TXWY. WX WAS MUCH LOWER THAN RPTED (VISIBILITY) AND AT MINIMUMS SAW APCH LIGHTS. CONTINUED TO 100 FT AND SAW RWY END IDENTIFIER LIGHTS. LNDG MADE AND SHORT ROLLOUT DUE TO AUTOBRAKES. TWR ASKED US TO CALL CLR ASAP WITH ANOTHER ACFT ON SHORT FINAL. WE SAW FOXTROT SIGN ON L SIDE OF RWY AT 60 KTS AND EXITED. CALLED CLRED AND THEN REALIZED WE TURNED OFF ON SIDE OF RWY AWAY FROM TERMINAL. MADE 180 DEG AT GND REQUEST AND CROSSED ACTIVE AFTER NEXT LNDG. TAXIED UNEVENTFULLY TO GATE. DUE TO LOW VISIBILITY; HURRIED APCH AND URGENCY IN TWR'S VOICE TO CLR; WE TURNED OFF AT FIRST VISIBLE EXIT. NORMALLY; WOULD OR SHOULD HAVE TURNED RIGHT; BUT THE R SIDE OF FOXTROT NOT VISIBLE; AS WAS LARGE YELLOW MARKER SIGN ON L SIDE. FEEL (AND A LEARNING EXPERIENCE) THAT DURING APCH CHK (BRIEFING) ON APCHS; ESPECIALLY DURING CAT II OR III; THAT THE RWY TURNOFF SHOULD ALSO BE BRIEFED; AS TO DIRECTION; R TURN; HIGHSPD; ETC. HAD LANDED ON PARALLEL RWY AT LEAST 6 TIMES IN LAST MONTH AND ALWAYS TURNED OFF L. LOW VISIBILITY; FEELING TO EXPEDITE EXITING RWY; ETC; LED ALL 3 CREW MEMBERS TO SEE FIRST AND MOST VISIBLE EXIT SIGN AND TURN OFF. COULD HAVE ROLLED ANOTHER 1000 FT TO R HIGHSPD. NO CONFLICT OCCURRED AND NO SAFETY FACTOR EVER DEVELOPED; BUT WE WERE NOT MENTALLY TUNED TO LOOK FOR AND THINK R TURNOFF. POSSIBLE DIFFERENT APCH BRIEFING MAY HAVE HELPED.