The Washington Post's On Faith page asks,
"Many in the West report that they incorporate Buddhist practices such as meditation or mindfulness into their own spiritual activities without necessarily adopting Buddhism as their religion. Can you practice Buddhism without becoming Buddhist?"
Oh, the weary task of defining terms.
The term "Buddhism" is an English language construction that, I am told, has no exact counterpart in any Asian language. Further, on the whole laypeople in Asia often don't become members of religious traditions, even when they observe them, in exactly the same way that westerners become members of, say, the Southern Baptist Convention or the Roman Catholic Church. So, I don't know if the English phrase "becoming Buddhist" exactly applies to anyone.
However, the English language is what it is, so we have to make the best of things. "Becoming Buddhist," to me, means taking the refuges.Very simply, this is a commitment to the Buddha's teaching as one's spiritual path. The path, or magga in Pali, is the path to realization of enlightenment and Nirvana. People who have made this commitment are also called "stream enterers."
(I realize that parts of the Nichiren tradition no longer formally take the refuges, but I suspect a sincere commitment to daily daimoku and gongyo amounts to the same thing.)
It seems to me that a person who, for example, practices mindfulness meditation only for self-improvement reasons is not a Buddhist. If there is no commitment to the Three Jewels, it's not what we in English crudely call "Buddhism." (See also the dharma seals.)
That said, I have no quarrel with people borrowing whatever they like from Buddhism and weaving it into another religious tradition or regimen of mental health hygiene. If it resonates with you, then go with it, and see where it takes you.