I found a term for a satisfying and fulfilling style of relationship that I engage in. It's called "casual relationship". I like the Wikipedia definition because it does not stress the sexual aspect.
So here's my definition, patched together from various sources and my own interpretation of it:
Casual relationship. A relationship with emotional and physical closeness, that may involve sexuality or may come close to sexual expression, when desired. There is no aspiration to long-term commitment and no expectation of exclusivity. The relationship may be part-time and does not dominate one's life and choices. It may be strong and intimate, but is intended to endure only so long as both parties wish it to. It includes mutual support, affection and enjoyment.
I read about this on the web and see biased language such as "motives for entering a casual relationship", or "because they are unwilling to commit to a full-fledged relationship" and "doesn't have time for a proper boyfriend". So I want to categorically declare that this type of relationship can be a legitimate relationship in its own right and not a transitional stage on the way to something else, or a compromise.