With the demise of The Hedge Fund Rule, you can look to Rule 203(b)(3)-1 to help you figure out how to count clients. The key part of the rule from a private investment fund perspective is (a)(2)(i):
A corporation, general partnership,limited partnership, limited liability company, trust (other than a trust referred to in paragraph (a)(1)(iv) of this section), or other legal organization (any of which are referred to hereinafter as a “legal organization”) to which you provide investment advice based on its investment objectives rather than the individual investment objectives of its shareholders, partners, limited partners, members, or beneficiaries (any of which are referred to hereinafter as an “owner”)
This rule is the opposite of The Hedge Fund Rule. Here the entity is the client, not the individuals or organizations in the entity.
For a private investment fund, the limited partnership is the client, not the invidiual limited partners.