The Christian god is the designer, builder, and ruler of the universe. It is believed to manage the day-to-day operations of the universe, at times by request (prayer), and at times using miracles. The first wave of scientists (Newton, Volta, Bernoulli, Boyle, etc.) explained how the universe operates according to regular laws which didn't need a ruler god, just a designer and builder god, and so, the deist god was created. It designed and built the universe, then left it.
The second wave of scientists beginning about the mid-19th century, demonstrated that the universe could assemble (build) itself without intelligent oversight (evolution, Big Bang). Matter naturally organized into filaments of galaxies of solar systems of chemical elements naturally, which then assembled itself into life then mind. Now, the builder god was no longer needed, and all that was left for a god to do was to create the initial state of the universe which then built and ran itself.
Today, the deist god is just a designer, not a builder or ruler. Maybe you're saying that that is not the deist god. Maybe it is a designer-builder god for you.
In my opinion, specified complexity refers to arbitrary substitution codes like human languages, where apart from onomatopoeia, words are purely arbitrary constructs with no inherent meaning, and their meaning can only be discerned by somebody who has learned the code (learned the language). Consciousness and intelligence are required.
Contrast that with the genetic code, where the association between codons (nucleotide triplets) and amino acids is made physically using transfer RNA. Here we have a snip of tRNA. Here, AGU (adenine-guanine-uracil) in the messenger RNA (mRNA) template pairs with a UCA anticodon which is mechanically tethered to an amino acid (serine). Using the human language analogy, the "word" AGU means serine the way cat means what it means.
Yes, that's an arbitrary association, but it doesn't need to be learned by the cell like a word in a human language must be learned. And when I say arbitrary in reference to the genetic code, I mean that AGU could have meant a different amino acid, but in that case, the DNA would have evolved different sequences to build this same protein. If nature had connected GGC to serine, then that sequence would need to appear in that location in the mRNA to build this same protein containing serine in this position. This, human language contains specified complexity which is arrived at by convention (agreement) and must be learned, but this code does not:
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Does language (spoken or written) indicate intelligent design to a goldfish? Probably not. It is unaware that these are written or spoken symbols with conventional meaning. It isn't experiencing the words as specified complexity. But spoken language can have natural meaning to anything that can hear. It means that there is a sound source nearby. It might also mean to run. But these are also hardwired meanings, not language in the sense of conveying thoughts. That must be learned. Smarter animals can learn to associate certain spoken words with their meanings, but not by convention or agreement, but by classical or operant conditioning, which is different from how human beings create words and assign them meanings.
And how does all of this tie into the threads opening question? At what point does a child hearing language realize that it was intelligently designed. Initially, it's learning like the other animals by induction (conditioning). The sound MAMA is passively associated with the qualities of a mother and the sound evokes the thought just like my dogs when I say, "treat." But eventually, that child will realize something that the dog never does - people can talk to one another using abstract and arbitrary symbols, and most will begin creating such symbols themselves as they give what they understand are arbitrary names to their stuffed animals, for example, names others won't associate with that toy until told what that name refers to. At this point, he has a glimmer of what specified complexity refers to even if unfamiliar with the phrase or concept explicitly.
And none of that addresses the matter of quantifying the specified complexity for the purpose of identifying it elsewhere. That seems as ephemeral as quantifying beauty or humor.