Design arguments for the existence of God are other minds arguments, i.e., arguments that other minds than one's own exist; this was explicitly recognized as early as Thomas Reid, who pretty clearly got the insight from Berkeley's arguments for God and for other minds, which are integrated with each other. The Argument from Analogy that serves as a starting point for almost every discussion of the problem of other minds (for instance, here and here) is a general form of the design argument discussed in Hume's Dialogues. Every kind of design argument also has a corresponding analogue for existence of minds other than God, so there is an analogy among different species in the genus. It's probably the case that there are other minds arguments with no analogue among design arguments, because design arguments are (among other things) a species of other minds arguments that involve some intermediary, and it's in principle possible that there are other minds arguments that don't -- but in general when we talk about other minds we are talking about human beings, and it is fairly clear that we know human minds through the intermediary of human bodies. We could even perhaps posit a spectrum of other minds arguments in light of the differences in intermediaries, ranging from a separate intermediary through greater degrees of integration of intermediary and mind to pure cases in which other minds are taken to be known directly and without intermediary:
(1) other minds via separated instrument
Separated instruments include anything separate from a mind that they nonetheless indicate, which makes for a very large field: computer, car, script, recorded vocal language, a corpse.
(2) other minds via conjoined instrument
The most obvious kinds of conjoined instrument are prostheses, but for our purposes we should also probably include a great many kinds of tools and instruments in active, ongoing use. A pen is a separated instrument from its designer, but a conjoined instrument for the one writing with it. According to Thomas Aquinas, angels can assume bodies but do not live them; if you met such a body and were trying to determine whether there was a mind to go with it, the body would be a conjoined instrument. Aquinas's Fifth Way would also fall here.
(3) other minds via organic instrument
The idea here is that we are dealing with something not identical to the mind but more closely integrated with it than a conjoined instrument; the mind inhabits, so to speak, the instrument. A pen in the hand is a conjoined instrument, but the hand is an organic one.
(4) other minds sans instrument
I've never run across a specific argument for an other mind that is both specifically designed to be of this kind and grown in the wild, but it's certainly possible to imagine what it might involve -- three obvious possibilities would be telepathy, innate ideas (possibly), and divine inspiration. On Aquinas's account of angelic knowledge, for instance, angels would know each other by divine inspiration and also by angelic speech (i.e., direct interaction of mind to mind). On Malebranche's account we know of God directly from God. (I say 'possibly' for innate ideas because I think it arguable that these are actually best understood as separated instruments -- e.g., Descartes's argument from the idea of God, which he does not, unlike Malebranche, regard as God Himself, analogizes ideas to machines.)
Given all of this, we can very well classify design arguments (any kind of design argument) in the same way:
(1) designer via separated instrument
What we most often call design arguments fall into this category. I see a watch, I infer there was a designer: the watch is a separated-instrument intermediary between me and the other mind that is the designer. When we are talking natural theology, these arguments are the ones that have a 'deistic' feel, although they need not actually require deism.
(2) designer via conjoined instrument
These are trickier to find; one often finds on closer analysis that candidates are really separated-instrument arguments. In natural theology, at least some arguments from miracles, from religious experiences, and from providential events would be examples. To borrow from and adapt Kant slightly, these are 'theistic' in character: they give us an designer who is not merely a cause of the intermediary but is acting or interacting through it. But, again, we need not be considering theism in particular. There are old vitalist arguments that are basically conjoined-instrument design arguments.
(3) designer via organic instrument
In natural theology these are 'pantheistic' in character, although, again, they need not actually require pantheism, or indeed any consideration of more than a small part of the world. Avatar-theophanies -- i.e., theophanies in which the god is the manifestation, such as one often finds in polytheism-- would be cases of organic instruments.
One can, of course, have arguments that are disjunctive by this classification; for instance, a design argument that does not depend on determining what kind of instrument is involved. And because intermediaries can be fairly different, none of this tells us much about the quality of any particular argument.
An interesting question, to which I have no clear answer, is how this way of thinking would impact on atheistic arguments from evil, which are 'nondesign' arguments. There tends to be a counterpart design argument for every argument from evil and a counterpart argument from evil for every design argument. So you could use this to classify arguments from evil. But it might well be a purely extrinsic classification.