In essence your question is about when a gender marker is needed and when it is just superfluous. Since Esperanto has only one gender marker, the female -in, this boils down to it.
I see this question more like a scale of tones than a strict on/off matter. While there still are people who use the female marker -in in titles like instruistino, pentristino, verkistino, they are in a clear minority. The closer to your immediate circle of relatives and friends you get, the more likely you make a distinction. Let me illustrate. Many already use amiko in a gender-neutral way (but not all as Neil Roberts points out in sia komento), but how many use kuzo that way? How about one's parents and siblings? Here I bet that very few use other than the gender-specific patro, patrino resp. frato, fratino. At least I count koramiko, koramikino into this innermost circle, since those words exist to denote that relationship in the first place.
When it comes to your other question, whether there is a gender-neutral term for that "special one", I doubt there is any official one, but you have to come up with a suitable one for that situation. Here are some possibilities (credits to Neil Roberts for some examples).
In a two-men or two-women relationship the persons are likely to define the other in terms of edzo, koramiko resp. edzino, koramikino.
There is the laŭfundamenta use of the prefix ge-:
- Marta kaj Johano estas gekoramikoj.
- Invitu viajn gekoramikojn al nia festo.
However in the latter example at least to me it is not so clear how many boy/girlfriends you have.
There is also nekutima use of the prefix ge-:
- Per amrendevua retejo oni povas trovi gekoramikon.
Some use this singular ge- to mark gender-neutrality despite what PMEG says about such use.
Whatever term you end up using, be prepared to explain it(*).
*: I worked as a volunteer during la 104-a UK and found that you need to explain a lot of things, things that are clear to you, but not for esperantists from other countries and continents.