It's not that surprising for the cuneiform spelling of proper names to sometimes vary, as they indeed seem to do here.
The first tablet (AO 5678) on the page you linked to is somewhat damaged, but line 7 is readable enough despite the damage and indeed appears to read UR.KIŠKI (𒌨 𒆧 𒆠):
Conveniently, the characters KIŠ and KI both appear undamaged in other lines on the same tablet, showing more clearly how the scribe normally wrote these signs and allowing easy reconstruction. For comparison, here's the entry for the character KIŠ in Labat's Manuel d'Épigraphie Akkadienne, showing how it was written in different periods and regions:
For what it's worth, CDLI also gives the same transliteration.
The second tablet (AO 19938), meanwhile, appears to read UR.KEŠ3KI (𒌨 𒋙𒀭 𒄲 𒆠; also affirmed by CDLI), where KEŠ3 is a composite character made up of the characters EN2 (𒋙𒀭, sometimes further decomposed as ŠU2.AN, although that appears to be a later reanalysis of what was originally a single character) and an otherwise rare character conventionally transcribed as HI×GAD (𒄲, i.e. HI = 𒄭 with a small GAD = 𒃰 in the center).
I say conventionally because, looking at Labat's entry for this character, the shape of the small "GAD" inside the square of the HI actually varies a lot — and the tablet appears to feature yet another variant not found in Labat's (admittedly limited) examples:
Given that this object is not actually a tablet, but rather a piece of limestone shaped and engraved to look like a cuneiform clay tablet, the graphemic variation might perhaps be attributed to a minor transcription error by the stoneworker, who was presumably working off an actual clay tablet prepared by a scribe, and might not even have been literate. Or, just as plausibly, it could also be an actual local variant of the sign.
(The missing vertical wedge in the "AN" part of EN2 = ŠU2.AN, which makes it look more like HAL (𒄬) rather than AN, is however featured in one of the variations of the sign documented by Labat above.)
It's perhaps worth noting that, as described by Labat in the entries shown above, both of these spellings echo the traditional written names of two ancient Sumerian cities, namely Kish (KIŠKI) and Kesh (KEŠ3KI) — or perhaps two traditional names of the same ancient city, if the theory that Kesh is just an alternative name for Kish turns out to be true.
Either way, any scribe with classical Akkadian scribal training would surely have been familiar with both names and, when asked to write down the city name "Urkesh", would have found it very natural and appropriate to choose a spelling that alludes to these ancient and prestigious names.
Ps. CDLI has traced drawings of both ARM 28, 044 and ARM 28, 046 but, alas, no transliteration for either. While the latter drawing does include several sign sequences that might read as UR.something.KI, there doesn't seem to be anything in either text that I can confidently recognize as either UR.KIŠKI or UR.KEŠ3KI, at least not without the help of a transliteration made by someone better at reading cuneiform than I am.
I also can't tell if the drawing for ARM 28, 044 on CDLI also includes "044bis" or not. In any case, a trip to the library might be needed.
Pps. I also ran a regex search of CDLI for "ur.keš/kiš.ki" with a pattern that should (hopefully) match all possible variants of the middle sign and any likely variations in transliteration.
It turns out there are indeed several results for both ur-kiški and ur-keš3ki, but most of them — with the exception of the two foundation tablets you already linked to, and copies of them — are Sumerian administrative tablets which (I assume) are actually referring to a person named Ur-Kish or Ur-Kesh (i.e. "man of Kish/Kesh"). I don't think these results can be used to say anything definite about how the city name you're asking about was written.