Pronunciation of Bank, Tank, etc.: Bay-nk, Ray-nk or Baen-k or Raen-k?

All of the above answers are confusing phonemic with phonetic transcriptions. Transcriptions like /bæŋk/ are broad phonemic representations. A transcription like [bɛɪŋk] (which is pretty standard for the US and about how I actually say it) is a narrow phonetic transcription (notice the brackets, not slashes). The Japanese dictionary of English is using a broad representation for two reasons:

1. Primarily, to avoid specifying excessive detail that is likely to confuse a learner -- they need to learn the phonemes first, and the allophonic rules later
2. To avoid having to commit to one particular dialect

Basically, before /ŋ/, Americans diphthongize and often raise front vowels. The words "kit" and "king" are broadly /kɪt/ and /kɪŋ/, but narrowly something like [kʰɪ̞ʔt] and [kʰɪiŋ]. In terms of just the vowels, the vowel in "kit" (and nearly everywhere else that "short i" occurs) is significantly lower than the cardinal IPA /ɪ/ sound. The vowel in "king" is a diphthong that begins higher than the vowel in "kit" and moves higher still.

BTW, the "Inland Northern" dialect is nowhere near Montana. It stretches along the major Rust Belt cities from approximately the Chicago metro area to the Rochester, NY metro area, covering areas that border Lake Michigan, Lake Erie and Lake Ontario, as well as most of the state of Michigan.