yup
Maybe we'd get that scene later in a flashback when he returns home and jin-ho comes knocking...
“Mom, I’m an S-Rank Hunter!”
but does he disappear with awesome speed after the phone call? and does the white tigers' master realize his power in the same way he does here?
Yes. I think it's almost word for word how white tiger said it.
So a difference from the manhwa and the light novel is that Jin Woo doesn't immediately disappear when he gets infront of the reporters. He answers a phone call from his sister and he gets called arrogant and uptight for ignoring the reporters and taking a phone call.