i was travelling, so could not answer, queries will try to answer on daily basis 

BELOW is DEFINITION of PFI

(72)
 
“public financial institution” means—

(i)
 
the Life Insurance Corporation of India, established under section 3 of the Life Insurance Corporation Act, 1956 (31 of 1956);

(ii)
 
the Infrastructure Development Finance Company Limited, referred to in clause (vi) of sub-section (1) of section 4A of the Companies Act, 1956 (1 of 1956) so repealed under section 465 of this Act;

(iii)
 
specified company referred to in the Unit Trust of India (Transfer of Undertaking and Repeal) Act, 2002 (58 of 2002);

(iv)
 
institutions notified by the Central Government under sub-section (2) of section 4A of the Companies Act, 1956 (1 of 1956) so repealed under section 465 of this Act;

(v)
 
such other institution as may be notified by the Central Government in consultation with the Reserve Bank of India:

 
 
Provided that no institution shall be so notified unless—

(A)
 
it has been established or constituted by or under any Central or State Act 68a[other than this Act or the previous company law]; or

(B)
 
not less than fifty-one per cent of the paid-up share capital is held or controlled by the Central Government or by any State Government or Governments or partly by the Central Government and partly by one or more State Governments;

Definition of Financial Institution

(39)
 
“financial institution” includes a scheduled bank, and any other financial institution defined or notified under the Reserve Bank of India Act, 1934 (2 of 1934);

NBFC will be covered in definition of financial institution but not public financial institution 

yes correct word is “Public Financial Institution”