What is the word for "understandable by specific group"?

There are lots of words covering the general concept, but they're usually used by people who are not members of the "specific group", so they tend to be pejorative.

Esoteric strikes me as somewhat less value-laden than most.

While not a direct answer to your question, the term jargon refers to language (words and phrases) that are common in one specific group but not used that way outside the group.

For example, in medical jargon, a heart attack is called a myocardial infarction.

An "in-joke" is a term used describe an event that is humourous only to a small group of insiders to whom the full history leading up to the event is known, thereby allowing them to find humour in the situation that would otherwise to an outsider not appear to be humourous.

You may be looking for the word shibboleth. From Wikipedia:

A shibboleth ( /ˈʃɪbəlɛθ/ or /ˈʃɪbələθ/) is any distinguishing practice that is indicative of one's social or regional origin.

It is also used to mean words used by a members of a group to distinguish themselves as members of that group. When used this way, it is being used metaphorically.