The use of the term "logotype" is fraught with confusion. In its broadest sense it would include all instances where a word ("logos") is cast on a single type. Actually, its use has been broader still, encompassing the casting of fragments of words as single types.
My perspective here is that of a typefounder and historian of composing machinery. In these fields the term "logotype" has a special association with various systems by which single types cast with multiple letters on the for the purpose of speeding hand composition. In such systems, the resulting printed text would look identical to the text if printed without the use of these logotypes; they are purely a compositional aid. So in my own use I'll reserve the term "logotype" to refer just to this kind of item. Various systems for logotypes, in this sense of the word, are discussed in ../../ Noncasting -> ../ Logotype Schemes.
There are, however, other cases in which multiple letters may be cast on a single type. In each of these cases the characteristic which distinguishes them from "logotypes" as defined above is that this combination is done for visual or aesthetic reasons. The resulting printed text does not look the same when printed with these (vs. plain single-letter types). I'll call these "combination" types. (This isn't a standard name for them; there is no standard name for them. I just made it up.) Examples of "combination types" include:
For the typefounder, the question of whether digraphs such as the Spanish 'll' are multiple single types or combination types depends, tautologically, on whether they are cut/cast as a unit for one type or not.
"Card Logotypes" and "Billhead Logotypes" in: MacKellar, Smiths & Jordon 1892 Specimens of Printing Types (where they occur on pp. 489 and 491).
For Catch Words see Specimens of Wood Type Manufactured by the Wm. H. Page Wood Type Co. of 1878
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