[SteveA]

The problem with 3 is that there's nothing to guarantee the relationship is desirable or subjectively beneficial.

Relationships 1 & 2 are, by definition, desirable by both/all parties involved. Relationship #3 avoids this by saying a non-desirable relationship can still be called "friendship" as long as it's "virtuous". Undesirable and virtuous seems to be conflicting, as I've assumed virtue is a desirable trait, so there's something wrong with the definitions, IMO.

#3 either isn't a friendship in the typical sense that both parties desire the relationship or virtue includes things that aren't beneficial to everyone. If true altruism is self-sacrifice for someone else, even to the point of not feeling any benefit is received, than maybe that qualifies but such events are rare, from my viewpoint, and it should be emphasized that either party is still free to terminate the relationship, even under the questionably virtuous form.