CathInfo.com

Q: What makes a person Catholic?
A: Baptism, with proper form, matter, and intent.

If someone says he/she is Catholic, he should be believed unless there is cause to question his baptism and it was obviously deficient in one or more of the three requirements. 

Within the large group of validly baptized Catholics, there are the devout, the careless, the educated, the ignorant, the holy, the evil, the practicing, the non-practicing...

Like it or not, Joe Biden and Nancy Peℓσѕι are Catholic. Especially given their ages, one can assume they were validly baptized. One can say, even prove, that they are bad Catholics.
Fr. Chazal, Bps. Williamson and Zendejas are Catholic, as is Fr. Pfeiffer of KY!  Words and actions tell me that the first three are good Catholics, and Fr. Pf. is a confused Catholic.
Within Catholicism are numerous distinctions, terms and definitions invented by men for differentiating, to see “who is on the same page,” so to speak.  This is something not dogma, not part of the Magisterium, not in Scripture.

To declare a person Catholic or not Catholic on the basis of a man made distinction is incorrect, and, I think, sinful.  It’s rash judgment.  Even the OP’s exclusion as Catholic, of those who attend TLM, “indultarians,” is wrong.
Look again at the reasoning and conclusion.  Is this not exactly why PF accuses “traditionalists” of being divisive, judgmental, and “rigid?”  It’s his justification for smashing the traditional mass and Sacraments. 
Nonetheless, it is God, not any man, is the judge of who is Catholic.  He will judge each Catholic on the basis of whether he lived a Catholic life—or not!

By that logic we get into the weeds of indifferentism. Because if simply valid baptism makes you a Catholic, then baptized Protestants, EO, atheists, Buddhists, Hindus, Moslems, etc. are all Catholics.

To be Catholic one must be validly baptized AND profess the Catholic faith.

By that logic we get into the weeds of indifferentism. Because if simply valid baptism makes you a Catholic, then baptized Protestants, EO, atheists, Buddhists, Hindus, Moslems, etc. are all Catholics.

To be Catholic one must be validly baptized AND profess the Catholic faith.

Technically speaking, a validly baptized Protestant IS Catholic.  I don’t don’t know what EO is, but atheists, Buddhists, Hindus, Muslims are NOT Catholic.  They do not baptize with water, “In the Name of the Father, and of the Son, and of the Holy Ghost.”  Many, but not ALL Protestants follow the Catholic formula.  If the intent is to bring the person into the true faith, and the person does not know the Catholic Church is the only true faith, ie. a material heretic, it is still Catholic.  A non-Catholic, non-“christian” can validly baptize in event of emergency.  The soul of the baptized becomes Catholic.  This is why lay-baptism should only be done if death appears imminent.  It should NOT be done if there is no assurance the person will, if recovered, continue in the Catholic Faith, or if a child below age of reason, not be raised Catholic.  Because they will be judged as a Catholic. Baptism confers a permanent mark on the soul. It IS what makes one Catholic. Consult a catechism. Profession, while meritorious, has no power to make a person Catholic.  The notion that one must make a profession of faith AND be baptized is a Protestant heresy introduced by Zwingli, (anabaptism), about 30 years after Luther.

One Universal Church of the Faithful

https://www.cathinfo.com/baptism-of-desire-and-feeneyism/one-universal-church-of-the-faithful/