Choice 1:
They consider themselves to conduct themselves 'morally' and they have perhaps 'high morals' or live a life they deem to be morally good often or at all times.
Choice 2:
They may not consider themselves to live 'morally' at all times or many times, but they contemplate the morality of many things and morality is a consideration for them in many decisions they make.
If you interpret both meanings then please choose whichever you feel is more highly implied or understood when you hear the phrase.
1,000When someone says someone else has a strong moral compass, I take that to mean that they feel strongly about doing the right thing. This does not necessarily imply that they "contemplate" morality for their decisions as much as they have a relatively sharp instinct for it.
If someone says that they themselves have a strong moral compass, I take that to mean that they probably meant choice 1 but don't actually live that life. Same thing applies to the self proclaimed rationalists IMO to give context.