Which is appropriate for an employee who does not fit anymore for the job, to file a resignation or to request for termination?

If you already have warned the employee then you may ask him to leave the job and again ask him for the resignation.

you're giving only two choices where there are other possible better choices. however, from the two you mentioned, it is always better to resign than get terminated as a reputation for you.

Resignation

If it has been decided to separate the employee with organization due to non-fit. Make sure this is not an abrupt decision; discuss the performance short-coming with the concern and his supervisor so that every-one is on same page.  The process for this is;

1 – Ask the employee to meet at some interview room or separate room where only HR and he is present.

2 – Quickly review the performance issue and expectation from boss and convince him to sign the resignation which indicate the reason most probably personal with immediate notice (HR must take printout of resignation in advance)

3 – In case of any resistance from him/her communicate them that this is the most feasible of closing the chapter, otherwise, termination letter could be used.

4 – Take the belonging from the employee and do handover him resignation acceptance so that for future reference he could use it.

5 – Ensure to process his cases on early basis to finance so that he’ll have full and final settlement and will focus on his career.

If the company have well prepared bad file about the employee so termination is the decision. But if not the resignation will be the solution to avoid complicated legel issues

Since termination is not that easy in some certain countries, why don't you think about rotating him to a different organization where he/she can fit for, I mean leave the termination action to the end if possible.