when experiments talk about cloning a person
Note there are no actual experiments about this, only thought experiments that exist specifically to explore peoples' reaction and intuition. There is no empirical value to them. Some of them take care to specify that "original" is meaningless (by splitting into 3 and destroying the "original", for instance), but people are pretty well used to thinking of "me" as one thing.
I prefer to define being same by causal flows.
Prefer whatever you like. I prefer not to use the word "same" in these cases - it's ambiguous and misleading. Specifying HOW they're alike and different is the only sane description, IMO. "fungible for a given purpose" is way more operationally sound than "same".
In thought experiments about cloning objects, people often call the object which maintained physical continuity "original" and other "clone". This seems like just a naming, but when experiments talk about cloning a person, people tend to care where their original version is. This feels dodgy in cases when person doesn't even feel their continuity (in sleep, for example), so I think that is a bit overvalued.
Let's take a black box cloning cars - perhaps taking the required set of details, correcting them on level of atoms to be indistinguishable from input car and constructing the second car. Inside the box, it is pretty obvious which car is original and which is a clone, but on the outside one can only learn that from box specification.
We can add an operation inside the box: let workers use a quantum RNG, and swap the result cars iff the random bit was 1. Then, person on the outside cannot tell which car is a clone. One could say "I don't know that, however there is 50% that cloned car is left one and 50% that it's right one instead"; I'd respond "the question is ill-formed since your definition of original doesn't really apply to this case".
I prefer to define being same by causal flows. For instance, let's open a window in the car before putting it into the black box and check which result car is causally downstream of input one - that is, has window opened. Once we see that both products have their windows opened, we can name both same[1], and same as the input.
that said, in future the cars might continue to be approximately same or very similar, or might not if one gets in a crash; their evolutions are different because of the outer world (drivers, for instance)