Regarding the "i" in "think" vs "bit"

Solution 1:

First, if you're actually teaching English to non-native speakers, you must learn and use at least those IPA symbols that represent English phonemes. Get yourself a copy of Kenyon and Knott and use it; or borrow one of your students' bilingual dictionaries. If you help them, your students can understand the pronunciations as they appear in their bilingual dictionaries, which always use IPA symbols; but they won't understand what you mean, otherwise. And they won't improve.

Second, the correct vowel symbol is /ɪ/, which appears in hill, shit, think, Mrs, and rim. We're talking about a high front lax vowel here, not a "short vowel", and certainly not about the letter I; calling something "short I" is 18th-Century nonsense. It's not short in Modern English, and it's not pronounced /ay/. How words are spelled in English has very little to do with how they're pronounced. English spelling is a good system for Middle English, but it's a bad system for Modern English.

Third, you're correct that a vowel before a nasal consonant /m, n, ŋ/ in English will sound different, because all vowels (including /ɪ/) that appear before nasals are nasalized. That is to say, they are pronounced with the nasal passage open, instead of closed.

This is because the nasal passage has to open in order to pronounce the nasal consonant, and the vowel anticipates this opening. This is a standard, predictable, utterly automatic, allophonic process in English phonology, and it does make the vowel sound different. But it doesn't make it a different vowel, because nasalized vowels occur in English only before nasal consonants, so they never contrast with non-nasalized vowels.

Finally, it's phonetics, not phonics. Phonics is a system of lying to English-speaking children in an effort to teach them how to spell their own language, and is useless for adults who don't already know English. (It's pretty useless for children, too, as we can see anywhere.)

Solution 2:

In California and the Southwestern U.S., the /ɪ/ in think is pronounced more like the vowel /i/ in bean, so it's close to /θiŋk/ in IPA. See this post on dialect blog, which calls it pre-velar tensing. In fact, I suspect this pronunciation is also present in the speech of some Americans who are not from the West, just judging from the way some speakers on forvo.com pronounce king.

The standard way to pronounce English is to use a short i as in /bɪt/ in these words, and this is the pronunciation you should teach to ESL learners. Your students are doing it correctly; don't uncorrect them.

UPDATE: Even if the difference you are noticing is between nasalized vowels before "ng", "nk" and "n" that John Lawler points out, and not this California pre-velar tensing, I don't believe it's the most urgent thing to correct in an ESL student ... these are two allophones of the same phoneme, and so should be perfectly understandable to a native English speaker (even if does produce a noticeable foreign accent).

Solution 3:

Nasal vowels may be a feature of US speech, but I’m not sure they occur in non-regional British speech. I hear no difference between the vowel in bit and the vowel in think and for the latter, indeed, the OED has /θɪŋk/, and not /θɪ̰ŋk/, for the US pronunciation. Nasal vowels are a noticeable feature of French, where they typically occur without a following nasal consonant. To be sure that they exist in English it would be helpful to be able to identify such sounds in isolation. Can speakers of American English pronounce the first two sounds of /θɪŋk/ alone as /θɪ̰/? (The tilde represents nasalization.)