Learn English – Phonics, the letter “I” and its rules

First, it's "shwa". It's a Hebrew word, not a German one, so there's no reason for SCH.

Second, it's both a phone [ə] in IPA, and a phoneme /ə/ in English. As a phone, it's got the sound of the final vowel in German Danke, of the first vowel in French Le Mans, or the first vowel in English the man. There is no shwa in Spanish or Italian.

Third, as a phoneme in English, /ə/ doesn't contrast with any other central vowel, so it has a lot of allophones: /ə/ [ɨ] [ə] [ʌ] (in increasing order of stress and decreasing order of speed), plus syllabic resonants [ṃ] [ṇ] [ḷ] [ṛ], before those consonants.

The best way I can suggest to practice the sound [ə] is to open your mouth to say an [e] (whatever that you think that is in your language), and then — while saying it, and without changing how your mouth or lips are positioned — move your tongue backwards toward the center of your mouth.

What you wind up saying is likely to be something close to a shwa.

Something funny happens to short i in some California accents; what most of us pronounce as short i (as in sit or king) turns into long e (as in seat or keen) when it's before an "nk" or an "ng". So ink would be pronounced eenk in these accents.

But this is a regional thing, established in California, Michigan, and probably several other regions of the U.S., but there are lots of regions where people don't do this. So the dictionaries are correct; except in California, pink and ring have a short i like bid.