OP: Does it imply that lay followers still engaging in the sexual act while monks giving it up are both following the Noble eightfold
path ? Or would there be yet a reason for such a distinction in
Yes, it is true that there is indeed a separation of the path for lay followers and for monks. The former is the version with defilements, and the latter is the noble version without defilements i.e. the full-fledged Noble Eightfold Path for monks (and nuns) only.
The path of the lay follower who practises Right Action, Right Speech, Right Thought and Right Livelihood as applicable to lay persons, is a valid path in Buddhism, even if it is not THE noble path that leads to the complete end of suffering. I speculate that the lay follower path may not go beyond stream entry, due to lack of renunciation.
From MN 117 (translated by Ven. Sujato) regarding Right Action:
And what is wrong action? Killing living creatures, stealing, and
sexual misconduct. This is wrong action.
And what is right action? Right action is twofold, I say. There is
right action that is accompanied by defilements, has the attributes of
good deeds, and ripens in attachment. And there is right action that
is noble, undefiled, transcendent, a factor of the path.
And what is right action that is accompanied by defilements, has the
attributes of good deeds, and ripens in attachment? Refraining from
killing living creatures, stealing, and sexual misconduct
(kāmesu micchācārā). This is
right action that is accompanied by defilements.
And what is right action that is noble, undefiled, transcendent, a
factor of the path? It’s the desisting, abstaining, abstinence, and
refraining from the three kinds of bad bodily conduct in one of noble
mind and undefiled mind, who possesses the noble path and develops the
noble path. This is right action that is noble.
The word "abrahmacariya" is not used above.
This sutta contains similar dual versions for Right Speech and Right Livelihood. Interestingly, there's also Right Thought (with two versions).
After writing the above answer, I found that a wise lay follower can become an Arahant on his death bed in SN 55.54:
If they reply, ‘I have done so,’ they should say: ‘Good sir, the
Brahmā realm is impermanent, not lasting, and included within
identity. It would be good to turn your mind away from the Brahmā
realm and apply it to the cessation of identity.’
If they reply, ‘I have done so,’ then there is no difference between a
lay follower whose mind is freed in this way and a mendicant whose
mind is freed from defilements; that is, between the freedom of one
and the other.”
Ven. Thanissaro's commentary to SN 55.54 indicates that a wise or discerning lay follower in this sutta, actually refers to a lay follower who has at least achieved stream entry.