In phonology, the "clean i" sound, IPA
/i/, is called "close front unrounded vowel", which, in Chinese is called:
The "dirty i" instead, as I learned it, should be the IPA
/ɨ/, which is called "close central unrounded vowel", in Chinese:
Those are the terms used in phonology, which are (more or less) scientifically validated.
For the "dirty i", the IPA
/ɿ/ has also been used, especially by sinologists, to describe the
-i sound with the retroflex initials "zh"/"ch"/"sh", but it's now considered obsolete:
Other characters have been added in for specific phonemes which do not possess a specific symbol in the IPA. Those studying modern Chinese phonology have used ⟨ɿ⟩ to represent the sound of -i in Pinyin hanzi which has been variously described as [ɨ], [ɹ̩], [z̩] or [ɯ].
As for colloquial terms, I'm not aware of any, because the pinyin syllables "zhi", "chi", "shi", "zi", "ci", "si", "ri" are said to be pronounced just as their respective initials (声母）“zh", "ch", "sh", "z", "c", "s", "r", except they are a little bit more loud and clear (响亮). Therefore the addition of the grapheme "i" after them is a romanization convention.
If you need to describe the term to someone who isn't familiar with phonology, or pinyin romanization, you can say it in your own words (or mine), but that's subjective. Example:
更响亮的舌面「衣」音 (衣 is simply used to represent the vowel sound)
A louder and clearer "i" sound, (pronounced with) the blade of the tongue (lateral sound).
A thinner and weaker "i" sound, (pronounced with) the tip of the tongue (apical sound)