"Luminous mind" in pali is pabhassara citta.
There will always be a problem when we start translating: citta, sometimes as consciousness, sometimes as mind; and there is another word in pali, vinnana, which is sometimes translated as mind, discriminating mind etc.
In my understanding of pali translation, translation of the use of a pali word is dependent on context, precedence and a whole host of other things.
There are opinions that to translate the cannon, understanding its context, the translator would have to have some experience of Buddhist meditation to get its flavour. If we say that is so, then how do we know that the translator is not incorporating his experience, basic or advance into his translation? If it is necessary and desirable then an advanced practitioner's translation would be much preferred, than to one who has no experience at all (if it is even possible to translate without any Buddhist meditative experience). We are after all talking about practice and experience.
Putting the problem of translation aside I'm going to decipher, IMHO, my understanding and translation of citta, vinnana and phabassara citta.
First I would quickly translate vinnana as mind or discriminating mind, and put that aside or away as not concerned with citta, which is translated as consciousness and phabassara citta, which would be luminous consciousness.
As we all know from our Abhidhamma there is only ONE consciousness (citta) and because of the many different mental contents (cetasikas) that arise with it we get different flavours of citta ( 89 flavours) etc...
Now what is this ONE citta?
And what is this arising and ceasing of the different flavours of citta, if it is not this one citta?
What does it mean for the citta to cease: does it mean to disappear into nothingness or disappear into something?
If it is nothingness, where does what makes up the citta and mental contents (call it energies, tendencies or whatever) goes and how does it arise again from its cessation? If its further arising is due to tendencies, where are these tendencies stored when the citta disappear?
If it is something, it must only be citta as there is nothing else that is other than citta.
It takes a lot of faith to believe that the citta disappears into nothingness and then arises again from nothingness almost intact with its kammic tendencies untouched.
What other traditions have explained (including the Thai forest Theravadin tradition, through the experiences of accomplished meditation masters like Ajahn mun) is that, that one citta that we have been talking about is the phabassara citta, "luminious consciousness/mind(loosely)" or primordial mind, clear light mind, buddha nature, bodhicitta, etc..
That the arising and ceasing of consciousness is "apparent". These flavours of consciousness arises and ceases within the "primodial consciousness" which is as yet, unseen, and seemed like nothingness. That is where avijja lies, where all the kammic tendencies are stored, the primordial consciousness.
How do we know it is there when it seemed like nothingness?
In a yet unenlightened being it is just "awareness". When the mind is empty it has this basic quality of "awareness and knowing" in an advanced practitioner and "awareness and knowing" has no shape or form?
In the end when we truly see it, it can be described as "consciousness without surface".
I've not found any description of soul that matches what I've said, so I'll pass it.
EDITS: (I've added further explanations in my comments with @Sankha below)
With a primordial consciousness, pabhassara citta ( which can be compared to an ocean with its waves), which is "stained" with kamma craving energies, creating the waves of these "apparent" arising and ceasing of the kusala/akusala/kiriya cittas, we continue to have the arising of the cuti (death) citta followed by the patisandhi (re-birth linking) cita.
At the end when craving is ended there is only the kiriya cittas waves in an Arahant's mind. When the last wave of the cuti citta ceased into the primordial consciousness, this primordial consciousness purified is the only permanent pure awareness, the nibanna without remainder, unestablished.
There is nothing that I've said that is not in accord with the Abhidhamma with only the addition that other traditions have added, that the ceasing is not into nothingness but into a primordial consciousness. What is the primordial consciousness in different traditions? Is it permanent?