"While it might sound impressive for people who do not know much about it, I don't like to discuss with someone who can't express a clear idea."

Did you recognize who wrote above sentence ?
It doesnt sound quite nice , does it?
Anyway let's put aside such childish ( from both parties ) disputes

Now you provided an explanation
However unless I completely misunderstood your words,in my opinion it is a wrong explanation

You said

"During flight, A thinks that B is aging slower while B thinks that A is aging slower "

and then

•When B turns around - however that is acheived - A ages rapidly when seen from frame B"

for " turns around " do you mean that B has reached the farthest waypoint and he is returning to earth ?

If so
First of all I dont get what you mean for " When B is flying "
B is alwayes flying
except of course at the departure and at the arrival
Consequently the former sentence is alwaye true , the latter alwayes false

However to get rid of the complication due to the "inertial frame " and similar stuff, put it this way
Suppose that the duration of the journey ,at costant speed, is very long , so long to make negligible the departure / arrival / change of direction times

If so , the simple formula of the theory of the special relativity is applicable with a great degree of accuracy, you get:

A proper time higher than B proper time

It seems unbielivable but it is like that