“Teaching experience” for visa purposes means paid employment as a teacher. Usually that would be full-time and usually professional teaching jobs come in multiples of years or semesters, at least. It doesn’t mean “I’ve taught some people informally” or “I did a short internship as part of my course” or anything like that. If you had teaching experience of the type they’re talking about, it should show up on your qualifications (actually, on your resume, backed up by contracts or other relevant documents.)
I know of teachers who had no problem getting a work permit with their 2-year college diploma + TESOL. You may want to print copies of the applicable regulations (Here are the Chinese and English versions)
Of course, as others have mentioned, YMMV depending on which clerk handles your paperwork. A copy of the law and a friendly smile may do wonders.
Your welcome as well.
My last recommendation is to give the Ministry of Education a quick call and run it by them.
It won’t stop every potential problem, but at least you can hear it from the cows mouth.
Good luck 
JDM-086, Steve4nLanguage navigated to the precise article in the relevant law and linked to it, so that the reader doesn’t have all the irrelevant text of the rest of the law. What Steve4nLanguage linked to is easy to print out and show someone. His post is two posts above this one.
The law to which Steve4nlanguage linked refers to “[f]oreigners to be employed as foreign language teachers as specified in Article 46.1.4 in the Act.” Just in case you show someone a printout of the law Steve4nlanguage linked to and they want to know what Article 46.1.4 of the (Employment Services) Act says, you can show them Article 46.1.4 of that law.
Here’s the Chinese version of Article 46 of the Employment Services Act:
laws.cla.gov.tw/Chi/FLAW/FLAWDOC … 128&lno=46
Here’s the English version: laws.cla.gov.tw/Eng/FLAW/FLAWDOC … 128&lno=46
The English version of the relevant part of Article 46 says:
[quote]Article 46 Unless otherwise provided for in the Act, the work that a foreign worker may be employed to engage in within the Republic of China is limited to the following:
- Full-time teacher teaching course(s) on foreign language(s) at a short-term class registered for supplementary schooling in accordance with the Supplementary Education Act. . . .[/quote]
The Chinese version of the relevant part says:
[quote]第 46 條
雇主聘僱外國人在中華民國境內從事之工作，除本法另有規定外，以下列
各款為限：
四、依補習教育法立案之短期補習班之專任外國語文教師。[/quote]
In my opinion, if you’ve been able to follow all the stuff posted in this thread, you’re well qualified to teach English here. 