The three systems, distinguished by one question: where does the first division happen?
Strict per stirpes divides the estate at the generation of the named children, whether or not any of them is still alive. California Probate Code section 246(a) states it exactly: the property "shall be divided into as many equal shares as there are living children of the designated ancestor, if any, and deceased children who leave issue then living," with each deceased child's share divided the same way among that child's descendants. The branch is fixed at the children's level and the size of a grandchild's inheritance depends entirely on how many siblings they have.
Division at the nearest surviving generation, sometimes called modern per stirpes or per capita with representation, starts one level lower when it has to. California Probate Code section 240, the state's intestate default, divides into as many shares as there are "living members of the nearest generation of issue then living and deceased members of that generation who leave issue then living." If any child survives, this produces the same answer as strict per stirpes. If no child survives, it does not: the first division happens at the grandchildren's generation and every grandchild takes an equal share regardless of family size.
Per capita at each generation takes the equality principle further. California Probate Code section 247(a) divides at the nearest generation with a living member, allocates one share to each living member there, and then provides that "the remaining shares, if any, are combined and then divided and allocated in the same manner among the remaining issue." Massachusetts General Laws chapter 190B section 2-106, titled "Representation," uses the identical "combined and then divided" mechanic, and Maine's corresponding section is titled outright "Per capita at each generation." The effect is that everyone in the same generation who inherits at all inherits equally, regardless of which branch they came from. This is the Uniform Probate Code's chosen system for intestacy.
Which phrase gets you which system is a matter of state drafting, and the states have not agreed. Massachusetts and Maine both apply the equalizing method when someone dies intestate, but both provide separately that a governing instrument calling for property to be distributed either "per stirpes" or "by representation" divides at the children's generation, which is strict per stirpes. So in those states a family with no will is treated one way and the same family with a beneficiary form is treated another. California reaches the same answer as those instrument rules for all three of "per stirpes," "by representation" and "by right of representation," but its intestate default is a third method again, the section 240 nearest-generation division. The practical rule that survives all of this: read the definition in the instrument first, then the governing state's statute, and never assume a phrase you have seen defined in one state means the same thing in another.
Beneficiary-form "per capita" is a fourth usage. On a retirement account or insurance beneficiary designation, the choice offered is often per stirpes or per capita, and there "per capita" typically means something narrower than any of the probate systems: a deceased beneficiary's share is redistributed among the surviving named beneficiaries, with nothing passing to the deceased beneficiary's descendants at all. That is a real and common option, and it is not what per capita means in a probate code. Forms also vary in what per stirpes options they offer and how they define them, so the definitions printed on the form itself are the operative ones.
The retirement-account consequence people miss. A per stirpes designation does not merely allocate dollars; it creates contingent beneficiaries who may become actual beneficiaries at your death. For a retirement account that raises a separate question with its own rules: whether those substituted takers count as beneficiaries for the required minimum distribution rules. Under Treasury regulation section 1.401(a)(9)-4, a beneficiary is taken into account if they were designated under the plan as of the date of death and none of the disregarding events has occurred by September 30 of the following year. The regulation also confirms that a beneficiary "need not be specified by name," provided they are identifiable under the designation, which is what makes a per stirpes instruction workable in the first place. So the two questions stack: per stirpes decides who gets the money, and the retirement distribution rules separately decide how fast they have to take it.