There are direct references to their friendship in all of the forms you have described: statements by Kirk, statements by Spock, and statements by other crew members (McCoy in particular).
Kirk calling Spock his friend:
KIRK: Spock is my best officer, and my friend.
AMANDA: I'm glad he has such a friend. It hasn't been easy on Spock. Neither human nor Vulcan. At home nowhere except Starfleet.
McCoy commenting that Spock is Kirk's best friend:
McCOY: Jim, the receptacles. Spock's consciousness was in one of them.
KIRK: It was necessary.
McCOY: What are you talking about? There is no Spock to return to his body. You've killed a loyal officer, your best friend.
McCoy commenting on Kirk and Spock being friends:
McCOY: These men are friends. To force them to fight until one of them is killed —
T'PAU: I can forgive such a display only once.
Kirk calling Spock his friend:
KIRK: I can't let Spock die, can I, Bones? And he will if we go to Altair. I owe him my life a dozen times over. Isn't that worth a career? He's my friend.
Kirk emphasizes it again later in the episode:
KIRK: Bones. He's my first officer and my friend. I disregarded Starfleet orders to bring him here.
Spock calling both Kirk and McCoy his friends:
SPOCK: It is my right. By tradition, the male is accompanied by his closest friends.
KIRK: Thank you, Mister Spock.
Spock calling Kirk his friend:
SPOCK: He does not know. I will do what I must, T'Pau, but not with him! His blood does not burn. He is my friend!
Once again:
T'PAU: Live long and prosper, Spock.
SPOCK: I shall do neither. I have killed my captain and my friend. Energize.

Above, you can see Spock's relief upon finding Kirk alive.