Whether the definition allows more or not, there are more specific terms that SHOULD be used to describe an individual. Feeling attraction to 3 but claiming to be BI(Which means TWO) is EXTREMELY misleading and nowhere near straightforward enough.
Any further and a person is classified as Trisexual or Polysexual. They will still fall under those classifications even if bi is also correct.
There are specified terms for more than two. They are just as valid. They should be used.
Whether they count as Bi or not they would be being very much too vague because by your(And the dictionary's) definition Bisexual is also Pansexual, Omnisexual, Trisexual, and Polysexual.
It's extremely important to be specific with something as complex and Gender or Sexuality, otherwise, you may give people the wrong idea and they may think you Don't or do possibly feel attraction towards their gender.
Basically. Saying you're Bi when you're actually Pan would be f**king confusing lol