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.
Best Answer
You’re right: one is [aɪ] but the other one, which is shorter and higher, is [ʌɪ].
This phenomenon is called, dubiously at best, Canadian raising. Most North Americans do this with that diphthong. Wikipedia writes:
It happens mostly before voiceless consonants, of which the /s/ at the end of dice is one. That makes dice come out as [dʌɪs] but die is just the unraised [daɪ]. Other pairs are tight/tide, writer/rider, and the two versions of high school.