Yes, you are right, this is yet another FALSE ETYMOLOGY of Alexander Hislop that the Society (and the WWCOG under Herbert W. Armstrong) adopted in order to support their unhistorical claim that all religion comes from ancient Babylon.
Easter has its Old English form as éastre, which Bede (Temp. Rat. xv.) derived from the name of the goddess Eostre whose festival was celebrated at the vernal equinox (the Northumbrian spelling was Éastre, in fact). The Old High German version is ôstara and the original Proto-Germanic form is *austrôn- "dawn" (with the suffix *-t(e)ro- "in the direction of", found also in Old Norse austr "east", Old High German óstar "eastern", and Old English éasterra "more to the east, eastern"), ultimately derived from the Proto-Indo-European root *awes- "to shine". This goddess is only found in the form *Aus-t(e)ro- in the Germanic languages.... in other Indo-European languages, the root is *Aus-ós, as in Greek Eós, Latin Auróra (both are goddesses of the Dawn in Greco-Roman mythology), while the *aus-t(e)ro root in Latin has the sense of "south" rather than "east", hence Latin auster "southerly wind' and austrális 'southern' (whence Australia). The simple root *awes also gave rise to Sanskrit uvása "they shone" and usar- "dawn", Doric Greek aós "dawn", Latin aurum "gold", Old Irish fair "sunrise," Welsh gwawr "aurora", Albanian agu "dawn," Latvian ausma "dawn," Lithuanian aushta "day break", and Tocharian A wäs "gold". The suffixed form *aus-t(e)ro meanwhile also gave rise to Sanskrit ušas-tara- 'eastern', Lithuanian austrìnis "northeast wind", and Latvian austra "dawn".
Now, the Babylonian goddess Ishtar, the Syro-Phoenecian Astarte, the Canaanite goddess Athtart and god Athtar, and the Hebrew Ashtoreth all derive from an altogether different root (< *`thtr), which is philologically unrelated to PIE *awes (particularly because of the initial `ayin). The etymological origin of *`thtr however is problematic. The clearest candidate is *`thr "irrigate", found in such forms as Arabic `aththarî "soil artificially irrigated", but Mark Smith points out that this word could have been derived from the god's name since Ishtar/Athtar/Astarte in either male or female form was a deity of fertility and irrigation. A better candidate may well be Indo-European *(Ha-)ster "star" (the *H is a laryngeal, corresponding to the `ayin of the Semitic name), borrowed into Semitic through contact with early Indo-Europeans (the Hittites and the Indo-Iranian Mitanni kingdom were second millennium BC Indo-Europeans in contact with Mesopotamians). PIE *(Ha-)ster gave rise to Hittite haster "star" (which attests the initial laryngeal), Sanskrit str- "star", Avestan starem, Armenian astl, Greek astér, astrón (whence "astronomy"), Latin stélla, Breton and Cornish sterenn, Old Norse staírnó, Old English steorra (whence English "star"), and Tocharian A s´re . This word also appears in South Semitic in Tigre `astär "heaven," Ge'ez `astar "sky", Amharic astär "star", and Bilin astär "sky". Note especially that Ishtar/Asthart/Astarte was an astral goddess, representing Venus or one of the guises of Venus, and the Sumerian equivalent of Ishtar was Inanna whose name may derive from Nin-anna "Lady of Heaven" (tho this may have been a folk etymology of a non-Sumerian name). There have been some attempts to trace both the Semitic and Indo-European forms to a Proto-Nostratic, but the evidence for this is weak since the cognates are not found elsewhere in Afro-Asiatic.
Basically, the two names look superficially similar but each have different etymological parentages. If anything, Ishtar was borrowed by the Babylonians, not the other way around.