What is the definition of Feistel Cipher?
According to Definition 7.81 given by Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone's Handbook of Applied Cryptography, 


  A Feistel cipher is an iterated cipher mapping a $2t$-bit plaintext $(L_0,R_0)$, for $t$-bit blocks $L_0$ and $R_0$, to a ciphertext $(R_r,L_r)$, through an $r$-round process where $r\ge1$.
  For $1\le i\le r$, round $i$ maps $(L_{i−1},R_{i−1})\xrightarrow{K_i} (L_i,R_i)$ as follows: $L_i=R_{i−1}$, $R_i=L_{i−1}\oplus f(R_{i−1},K_i)$, where each subkey $K_i$ is derived from the cipher key $K$.
  Typically in a Feistel cipher, $r\ge3$ and often is even. The Feistel structure specifically orders the ciphertext output as $(R_r,L_r)$ rather than $(L_r,R_r)$; the blocks are exchanged from their usual order after the last round. Decryption is thereby achieved using the same $r$-round process but with subkeys used in reverse order (...)


On the next page they state, with the contradiction pointed in the question's Q1:


  DES is a Feistel cipher (...)
  An initial bit permutation (IP) precedes the first round; following the last round, the left and right halves are exchanged and, finally, the resulting string is bit-permuted by the inverse of IP.




In Q1: DES is a Feistel cipher, except for IP and IP-1; that deviation keeps the notable property of a pure Feistel cipher that only the order of subkeys differs for encryption and decryption.

In Q2: the construction is a Feistel cipher, except for the above deviation, and an additional final XOR with $K_2$ on encryption, and correspondingly an additional initial XOR with $K_2$ on decryption; this looses the aforementioned notable property of a pure Feistel cipher, and adds a key-dependent transformation.

I would say the answer to Q1 is yes for Jim Kirk, and no for Mr. Spock; while they could agree on no for Q2. The Captain is on option 3 in the question (perhaps also requiring that any key-dependent transformation is thru the Feistel structure); while the logician sticks to 2 (and wants any deviation to be explicit). Nobody on the deck stands for 1.