To qualify for the lower resident withholding rate, you have to be physically present in Taiwan for at least 183 days in a calendar year. Until qualifying, your withholding will be done at 20%. If during your first calendar year in Taiwan you are not present for at least 183 days, then your qualifying period will start over again on January 1. For any year you are in the country for at least 183 days, you will also qualify for lower withholding the following year as well.
Days physically present in Taiwan does not depend on your visa status, work permit status, or whether you have a resident card. You can enter on any kind of visa and those days still count toward your 183 days requirement. You can arrive and depart as often as you want, and each qualifying day counts (i.e. your 183 days doesn't have to be consecutive).
The way qualifying days are counted:
1) Your first day arriving in Taiwan does not count (except in case 4 below).
2) Any full day in Taiwan counts as 1 day.
3) Your day of departure counts as 1 day.
4) If you arrive and depart in the same day, it counts as 1 day.
If you are not sure, you can ask the foreign affairs police office or foreign tax office to provide a tally of your days physically present.
John enters Taiwan in February and studies Chinese for 6 months, and has already been in the country more than 183 days when he gets hired as an English teacher. His withholding should be done at the lower resident rate, because his time studying Chinese counts towards the requirement.
Mary arrives in September to start teaching English. She's puzzled when she's passed 183 days and her withholding is still done at 20%. Unfortunately, the qualifying period must be within the same calendar year. Because she was here less than 183 days the first year, her qualifying period resets in January and she wil need to wait until July to get the lower rate.
Martin has been in Taiwan for a couple of years and plans to go home in April. Since he will be here less than 183 days in his last year, what should the withholding be? Because he had been in the country more than 183 days the previous year, he still qualifies for the lower rate.
Edward has been in Taiwan for a couple of years. His work permit and ARC end in mid-June, slightly before the 183 days qualifying period. Unlike Martin, Edward plans to come back to work in Taiwan, but not until the following year. He's worried that when he comes back he'll be stuck at the higher withholding rate again. He is very close to the 183 days but he can't get an extension to stay long enough. On his last day he flies to Hong Kong and returns on a tourist visa and stays until mid-July. He now has stayed more than 183 days, and has requalified for the lower withholding rate. Being on a tourist visa still counts. When he returns in February of the following year, he will still qualify for withholding at the lower resident rate.
Please note that your withholding rate does not necessarily correspond to your actual tax rate. If you qualify for the lower resident tax rate within the calendar year, then all of your income will be taxed at the lower resident tax rate, even though the first 183 days of income was withheld at 20%.