What is the legal test for making a person an indispensable party?

In Barney v. Baltimore, 73 U.S. (6 Wall.) 280, 284, 18 L.Ed. 825, it is said: "There is a class of persons having such relations to the matter in controversy, merely formal or otherwise, that while they may be called proper parties, the court will take no account of the omission to make them parties. There is another class of persons whose relations to the suit are such, that if their interest and their absence are formally brought to the attention of the court, it will require them to be made parties if within its jurisdiction, before deciding the case. But if this cannot be done, it will proceed to administer such relief as may be in its power, between the parties before it. And there is a third class, whose interests in the subject-matter of the suit, and in the relief sought, are so bound up with that of the other parties, that their legal presence as parties to the proceeding is an absolute necessity, without which the court cannot proceed. In such cases the court refuses to entertain the suit, when these parties cannot be subjected to the jurisdiction."

In Williams v. Bankhead, 86 U.S. (19 Wall.) 563, 571, 22 L.Ed. 184, it is said: "The true distinction appears to be as follows: First. Where a person will be directly affected by a decree, he is an indispensable party, unless the parties are too numerous to be brought before the court, when the case is subject to a special rule. Secondly. Where a person is interested in the controversy, but will not be directly affected by a decree made in his absence, he is not an indispensable party, but he should be made a party if possible, and the court will not proceed to a decree without him if he can be reached. Thirdly. Where he is not interested in the controversy between the immediate litigants, but has an interest in the subject-matter which may be conveniently settled in the suit, and thereby prevent further litigation, he may be a party or not, at the option of the complainant."