Light relay turn on with sunset and turn off with sunrise

@Costas
yes, TZ range is from +14 to -12 in this sketch ,and the problem is in -negative Region.

Start at sunset
Stop at sunrise
Time zone: Pacific/Kiritimati
Time zone offset: 14 Hour(s)       --->   Positive  TZ  (+14)
Day 1 is selected
Day 7 is selected

Start at sunset
Stop at sunrise
Time zone: US/Samoa
Time zone offset: 1193035 Hour(s)      ---> Negative  TZ    (-12)
Day 1 is selected
Day 7 is selected

also in Nick Gammon’s Sunrise Sunset sketch 4 inputs are required and they all affect the result.

  /* TimeLord Object Initialization */
  myLord.TimeZone(TIMEZONE * 60);
  myLord.Position(LATITUDE, LONGITUDE);
  myLord.DstRules(3,2,11,1,60); // DST Rules for USA
 
 byte today[] = {  0, 0, 12, 6, 3, 2017    }; // store today's date (at noon) in an array for TimeLord to use
   /*  second 0, minute 1, hour 2, day 3, month 4, year 5 */

about your sketch with Sunrise and Sunset I got same result for all Time Zones.
" Motor STARTED at 1193046:28"
" Motor STOPPED at 1193046:28"

Checked schedule at: 23:59:48
ToDAY   Is   : 06/03/2017
Schedule ACTIVE today
Time zone: Europe/Nicosia         --->   here    Nicosia   in  Europe
Time zone offset: 2 Hour(s)
Day 1 is selected
Day 7 is selected
Motor STARTED at 1193046:28
Motor STOPPED at 1193046:28

Checked schedule at: 00:08:47
ToDAY   Is   : 07/03/2017
Schedule ACTIVE today
Time zone: Asia/Nicosia         --->   here    Nicosia   in  Asia
Time zone offset: 2 Hour(s)
Day 1 is selected
Day 2 is selected
Day 7 is selected
Motor STARTED at 1193046:28
Motor STOPPED at 1193046:28