One can claim that in the late thirteenth and early fourteenth centuries two different Trinitarian traditions developed in the west. The first, grounded in the thought of Thomas Aquinas, claimed that relations themselves individuate the persons of the Trinity. On this view, the Father is the Father because he bears a particular relation to the Son. The second, developing in and through the work of Duns Scotus and others, claims an emanationalist model where origins individuate the persons of the trinity. The person of the Father is innascible, and by virtue of this innascibility, is the Father. On the first model, bare relations individuate the Trinitarian persons. According to the second model, particular monadic properties are possessed by the persons, properties, on the basis of which, relations among the persons are established.
With regard to the Son being the Word, the first view, because it is committed to individuation through bare relations, understands 'Word' metaphorically. It is because that the Son bares sonship to the Father that the Son is the Son. Word can be spoken of this Son metaphorically because the Son is "like a Word" in being an expression (issue) of the one who expresses (issues) it. On the other hand, the emanationalist view can regard 'Word' as specifying more or less the individuating property of the Second Person. Because the Second Person is Word, He is called the Son of God.
The distinction between internal and external relations may be helpful here in understanding the distinction. An external relation R holds of a with respect to b if and only if R does not change the entity which a is in relation to b. For example, I am externally related to my son because I remain exactly who and what I am genetically regardless of whether the relation of being the parent of b obtains or not. An internal relation R' holds of a with respect to b, however, if and only if R obtaining changes what a is in relation to b. For example, my son is internally related to me because his genetic material varies as a function of whether the relation 'being a son of me' holds or not. An external relation does not change the being of the one related to another; an internal relation does change the being of the one related to the other.
These categories can be helping for thinking about the indiviudation of persons within the Trinity. On the Fransican model, the Trinitarian persons are externally related with respect to 'being the Father of x' because the dyadic relation of Father to Son is grounded on the the existence of nonrelational, mondadic properties like innascability which themselves individuate the persons. According to the Dominican model, however, the persons are internally related with respect to dyadic relational properties like 'being the Father of x', for the Father could not be the person He is apart from the relation in question.
Using the language of logic, the Dominicans claim that the one called 'Father' is the one having two dyadic relations: 'being the Father of the Son' and 'being the one from whom the Spirit proceeds'. For the Fransicans, on the other hand, the one called 'Father' is the one having the nonrelational monadic quality of innascability, a property, on the basis of which, the relation of Father to Son is established. The two dyadic relations individuate the Father on the Dominican model; a single monadic property individuates the Father on the Fransican view.
Now things get difficult when trying to conceive how either account is possible without compromising the unity of the divine essence, or without denying real objective distinctiveness of the persons. For instance, if we say that there are either real relations or real intrinsic qualities in the persons, we seem to be claiming that there is a real ontological consitutent to individuation, a modus essendi, not merely modi intelligendi. But if this is so, it appears that divine simplicity, and hence the divine essence itself, is compromised. On the other hand, if we say that the Trinitarian individuaters are mere modi intelligendi, then the ontological distinctiveness of the Trinitarian persons seems to be undermined. There is one divine essence ontologically, although human beings might, from the standpoint of reason, distinguish that essence into persons. But if this is so, then the Persons seem to be due to human construction.
The third way also sometimes used in the tradition appears unworkable. It claimed a modus se habendi in which no distinction arises when comparing the personal properties to the divine essence, yet a distinction arises when comparing these properties to the properties of the other persons. Lamentably, this is a non-starter because the problem still remains: Are the individuating properties, upon which the persons are individuated, ontologically grounded or not? If yes, then divine simplicity seems difficult to sustain; if no, then divine simplicity is sustained at the expense of their being any real differences among the Trinitarian persons.
The problem of the identity in difference of the Trinitarian persons is a profound one, and it is clearly not solved either in the Dominican or Franciscan traditions.