"Gaiden" contains the kanji for "outside" or "other" (外) and the kanji for conveying something as in a story or a legend (伝). So a more direct translation is "outside story". I've also seen it translated as a "story other than the legend" which sounds clunky but is accurate. A "side story" for example is a story that exists "outside" the bounds of the main story. For example, Fire Emblem gaiden it set in the same as the first Fire Emblem but because it takes place on a different continent it has no relevance to the story of that game. However, because the above mentioned phrasing allows for several possible interpretations, there isn't one specific translation that is always correct. Gaiden can also mean "prequel", "sequel", "prologue" "epilogue", or "spin-off". It can also mean "another story" which is a term directly borrowed from English and has the connotation of being a story that takes place in another world from that of the main story.
Plus, is that an advertisement (aka actual marketing) or a magazine review (aka not actual marketing)?
Japanese magazine reviews have been known to exaggerate and even just plain make shit up.
It's an advertisement from the back of a strategy guide appendix that came with bundled with an issue of "Gekkan PC Engine". The publisher of this magazine is the same one that published the actual official guide book, and the strategy guide appendix was intented as a preview for the official guide. Therefore, it's very likely the information it contains came directly from Konami.
Speaking of magazines, there is one I actually wanted to bring up. It's called MegaDriveFAN. It had a scope about the game's reveal, it ran a design contest for the game, and it later also contained an official guide. This magazine definitely had close ties with Konami, and therefore I think it's save to regard the information it contains as reliable.
The following issue is from July 1993, the issue that first unveiled the game and ran the design contest:
In this article, the game is described as "Konami's newest horror-action series". This is why the game is called "Vampire Killer" in Japan and not "Akumajou Dracula". Furthermore, it states that Vampire Killer is a new vampire-themed action game series. It is said specifically that Vampire Killer isn't a Akumajo Dracula sequel, and that it's a new game. So you can't interpret "gaiden" to mean spin-off or sequel in this case.
It also says that Vampire Killer and the Akumajou Dracula series share the same "Dracula theme". So what I think what it meant with "Dracula gaiden" in that scan I posted before is that Vampire Killer is also a "legend" about Dracula but it's different from the legend that is featured in Konami's other series: Akumajo Dracula. To use the same words as in the advertisement, with Vampire Killer "a new Dracula legend begins".
I think this "new Dracula Legend" was supposed to be different save for a few shared elements like Dracula and the main character being a whip-wielder. I think this also explains why the developers all of a sudden tied the game's plot to the events of Bram Stoker's Dracula and why they replaced the Belmonts with the Morris family. Would this have been a normal Akumajou Dracula project, I doubt such liberties would have taken since it would have limited what developers could have done with the story later. As we could have observed a few years ago, IGA was struggling enough as it is with trying to properly connect the game with the rest of the series.