Part 141 initial for an MEI

I’m a CFI/CFII and MEI. I received my initial CFI 5 years ago and completed my MEI several months ago. I work part time for a 61/141 school and I’m trying to ensure I’m legal to teach a student for their initial instructor rating which would be MEI. 

I’ve read the applicable FAR which is posted below. paragraph 3 applies because this is 141. My confusion is I believe as long as I’ve had my CFI for 2 years and have given 200 hours of instruction then I meet the requirements regardless if it’s in a single or twin aircraft. Should not matter that my MEI is only 2 months old. Correct?

Thanks in advance

(h) Qualifications of the flight instructor for training first-time flight instructor applicants. 
(1) The ground training provided to an initial applicant for a flight instructor certificate must be given by an authorized instructor who—
(i) Holds a current ground or flight instructor certificate with the appropriate rating, has held that certificate for at least 24 months, and has given at least 40 hours of ground training; or
(ii) Holds a current ground or flight instructor certificate with the appropriate rating, and has given at least 100 hours of ground training in an FAA-approved course.
(2) Except for an instructor who meets the requirements of paragraph (h)(3)(ii) of this section, a flight instructor who provides training to an initial applicant for a flight instructor certificate must—
(i) Meet the eligibility requirements prescribed in §61.183 of this part;
(ii) Hold the appropriate flight instructor certificate and rating;
(iii) Have held a flight instructor certificate for at least 24 months;
(iv) For training in preparation for an airplane, rotorcraft, or powered-lift rating, have given at least 200 hours of flight training as a flight instructor; and
(v) For training in preparation for a glider rating, have given at least 80 hours of flight training as a flight instructor.
(3) A flight instructor who serves as a flight instructor in an FAA-approved course for the issuance of a flight instructor rating must hold a current flight instructor certificate with the appropriate rating and pass the required initial and recurrent flight instructor proficiency tests, in accordance with the requirements of the part under which the FAA-approved course is conducted, and must—
(i) Meet the requirements of paragraph (h)(2) of this section; or
(ii) Have trained and endorsed at least five applicants for a practical test for a pilot certificate, flight instructor certificate, ground instructor certificate, or an additional rating, and at least 80 percent of those applicants passed that test on their first attempt; and
(A) Given at least 400 hours of flight training as a flight instructor for training in an airplane, a rotorcraft, or for a powered-lift rating; or
(B) Given at least 100 hours of flight training as a flight instructor, for training in a glider rating.

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