No; it can be any instructor "authorised" to do the training - if outside the USA.
Stated where? 61.41 does not state that such a person qualifies to be called an FAA "Authorized Instructor" in accordance with FAR 61.1(2).
Has the definition for training done outside the USA, which the FAA does accept, changed, to specifically say the instructor must be an FAA CFI/CFII?
Again, the relevant regs are 61.1(2) which gives the definition of the term "Authorized Instructor" which is later used as the term in the subsequent regs.
61.41(b) only refers to a foreign instructor making entries to show training given. It does not state that person qualifies as a FAA "Authorized Instructor".
61.109 states that 40 hours of flight time is required of which 20 hours must be with an FAA "Authorized Instructor". Thereby suggesting that hours outside of those 20 can be done elsewhere. A foreign instructor cannot sign of a student for a FAA PTS which is the purpose of the final 3 hours from the total of 20.
So reading between the lines would suggest that some hours can be done with a foreign instructor but at least 20 must be with an FAA "Authorized Instructor" (as defined in 61.1(2) ) including the final 3 hours required to be approved for PTS - solely based on what is written.
Like I said a FAA Legal Counsel clarification is what is really needed and so far I have not found one.
It is an interesting topic and open for interpretation based on which side of the fence you are standing.
Maybe B2N2 has some experience of this.