I'm going to weigh in as a therapist.
Fantasies are a great indicator of sexual attraction/orientation.
Sexual orientation is extremely complex, more so for women. The evidence is that men, after adolescence, are pretty much attracted to one or the other. Heterosexual men may engage in homosexual acts, but that is generally when they don't have a potential female partner (i.e. prisoners). Efforts, mainly funded by conservative religious organizations, to change men's sexual orientation haven't been very promising so far. So far, the evidence has been that getting homosexual men to participate in heterosexual acts doesn't change their fundamental desires. They still have more of a physical response to homosexual media.
There appear to be women who are only attracted to women; who have never had fantasies/attraction for men/boys. However, evidence has shown that MOST women can be attracted to either sex. For most women, it appears to be more a matter of conditioning.
If a therapist tries to tell you who to have sex with, fire him. He or she should help you understand your values and needs and the implications of your decisions. Even if you should have a pleasurable sexual experience with a woman, that doesn't necessarily prove anything. It isn't at all uncommon for women to "switch" to relationships with women after abusive relationships with men. That supports the concept that it is largely a matter of conditioning. Therefore, there is no absolute "test" to determine if you're a "lesbian" or not.
At this point, let's ask what value there is in applying a label to your sexuality. Only you know what pleases and excites you. You're clearly not "asexual." Sexual drive can be lowered by countless factors. These include depression, worry, poor nutrition, hormonal imbalances, relationship problems, .... Another possibility is that you're comparing your sex drive to other people and you're worried because they seem to have more desires. Just because your sexual needs and drives differ from a hypothetical normal, that doesn't mean that you're "abnormal" or that there is necessarily a problem. If you're saying "I'm not myself; what's wrong with me?" then it could be time to see a therapist.
Be wary of getting a therapist with an agenda. A therapist should work within the context of your values and goals. Sometimes, therapists define normal as "like me." That isn't always the case. "Normal" covers a broad territory.
I would encourage you to stop trying to fit yourself into a category and begin exploring your desires. One good way is to read books by women who are attracted only to women or bisexual. See if their descriptions match what you're thinking and feeling. A therapist can help you explore these questions, but he or she can't answer them for you. I think exploring the written thoughts and feelings of women with similar experiences can be a great place to start. Then, you'll know what topics you want to take up with a therapist.