You are out to gay strangers and medical personnel but not out to the most important people in your daily life - so yes, you are closeted.
Whether coming out is "easy" is irrelevant. It's a process that takes years (and, in heteronormative society, never really ends). You learn to be closeted, so unlearning takes about as much time.
Equating being out with "broadcasting" your sexuality is bizarre - and one tiny internalized step away from the homophobia implied by the term "flaunting." (And sex is not the sole dimension.) Being out is being you, not broadcasting or flaunting you.