The process of converting the letters of one script into the letters of another is called "transliteration"
Definition of TRANSLITERATE
:to represent or spell in the characters of another alphabet
— transliteration [...] noun
Indeed the scheme used in your OP is called IAST (International Alphabet for Sanskrit Transliteration).
You should ask for a "transliteration" or a transliterated version.
Now, for sure, sarvasya chaahaN hridi ... is also a transliteration, but it's not an academic/scientific transliteration. Both of these words ('academic' and 'scientific') are used for precise, scholarly transliterations of Cyrillic and I see no reason why they cannot be used more generally. If you want a precise, scholarly transliteration, what you must really do is specify the scheme, which is more important. There is nothing inherently wrong with the Indian government-approved Hunterian transliteration which uses few diacritics, or Harvard-Kyoto which uses its own schemes with no diacritics at all, but also doesn't strive to reflect pronunciation closely. So what you must do is specify the transliteration scheme which you are using. For example, IAST will transliterate ए as e whereas ISO15919 will use ē. Both, however, will transliterate आ as ā. Many transliteration schemes use diacritics. You can ask for a transliteration with diacritics, but you must specify the scheme, because different schemes may use different diacritics or use diacritics for different purposes.