The way I see it they are the same. The concept consists of a transaction in two parts, one between Alice and Bob, and on between Bob and Charlie.
Part 1: Alice helps Bob, and in lieu of payment says "pay if forward".
Part 2: Bob helps Charlie, fulfilling his "debt" to Alice. If Charlie offers to pay Bob, and Bob says "pay it forward", this begins a second transaction.
If we look at the internal thoughts of Bob during Part 2, they may very well look like "I was helped when I started out and now it’s my turn to help others.".
The OP describes the same thing with one minor variant: instead of Alice or Bob explicitly stating "pay it forward" the general rules of their culture have a built-in, default structure of "pay forward any karmic[1] debt". It is pretty easy to envision that after enough explicit statements of "pay it forward" that the statement need not be uttered because it is understood at that point.
Part 1: Alice helps Bob, and in lieu of payment says "pay if forward".
Part 2: Bob helps Charlie, fulfilling his "debt" to Alice. If Charlie offers to pay Bob, and Bob says "pay it forward", this begins a second transaction.
If we look at the internal thoughts of Bob during Part 2, they may very well look like "I was helped when I started out and now it’s my turn to help others.".
The OP describes the same thing with one minor variant: instead of Alice or Bob explicitly stating "pay it forward" the general rules of their culture have a built-in, default structure of "pay forward any karmic[1] debt". It is pretty easy to envision that after enough explicit statements of "pay it forward" that the statement need not be uttered because it is understood at that point.
[1] In colloquial sense of the term karma.