Those Ideas which are rais’d in the Mind upon the presence of external Objects, and their acting upon our Bodys, are call’d Sensations. We find that the Mind in such Cases is passive, and has not Power directly to prevent the Perception or Idea, or to vary it at its Reception, as long as we continue our Bodys in a state fit to be acted upon by the external Object.
Families of these sensations linked by resemblance, he goes on to say, are attributed to unified powers of receiving them, which we call 'senses'. The distinction between external and internal senses, Hutcheson takes to be a non-essential issue; it is based on a point that Spicher notes, namely, that they seem in some way linked to senses like sight and hearing, although not reducible to them because they can generate the relevant ideas in matters not involving them.
Thus Hutcheson's definition of a sense is a power that originates an idea to the reception of which the mind is passive, and the distinction between internal and external senses is a matter of convenience based on the relations of senses to each other. The idea of beauty is the kind of idea that designates a sense, and it is distinguishable from senses like sight, hearing, etc.; and this idea of beauty applies to things exhibiting Uniformity amidst Variety. Nothing here seems obscure. Conceivably Spicher could mean that Hutcheson does not give a mechanism for it, but this is not the natural way to read the claim.
Later he says, in discussing Gerard:
Gerard divided up his study into seven principles of the internal sense (or powers of the imagination), not only a sense of beauty like Hutcheson. The seven principles are novelty, sublimity, beauty, imitation, harmony, oddity (humorousness), and virtue.
But Hutcheson himself distinguishes the senses of grandeur (=sublimity) and novelty from the sense of beauty; he has another treatise on the sense of virtue. Harmony is a more borderline case, since Hutcheson sometimes speaks of the 'sense of beauty and harmony' as if it were one thing, and sometimes speaks of the sense of beauty and the sense of harmony as if they were distinguishable. Perhaps the idea is that Hutcheson's study focuses only on beauty (so that, for instance, grandeur and novelty are mentioned only to be set aside as not the topic under discussion)? But this seems a misleading way to put it.