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Just want to chime in here because I was fascinated by this PS1 game when I was much younger and did quite a bit of research on this. You're partially right about what LSD stands for, though the FMV you referenced is only one of many that could play during the game's opening sequence. I'm about to drop quite a bit of info and speculation here, but if you're impatient the TL;DR is at the bottom.
Osamu Sato (developer of the original PS1 game) has publicly stressed that this game takes its inspiration solely from the dream journal written by Hiroko Nishikawa and has no link at all to drug use. However, it is strongly theorized that the game's similarities to the experience of tripping on acid may have played a pretty substantial role in settling on the name. Sato may have also suspected (or maybe even known) that Nishikawa was under the influence of LSD when writing about his dreams. Take a look at the original Japanese website for LSD via the Wayback Machine. Here's the (google translated) text at the top: "WHAT IS L.S.D.? ... It's not a game of acid eaters and glimpses of new dimensions. Taking the acronym 'Link', 'Speed' and 'Dream', 'L.S.D.'" If you read the rest of the commentary there, the developer says that he doesn't condone LSD use, but goes on to say that if no one finds out about it, you won't get caught. Take from that what you will.
The game started off with the title "Link Speed Dream" to describe the gameplay's method of linking dreams by walking into objects, but over the course of development, the title morphed into something more ambiguous. Ultimately, I think the acronym's ambiguity was probably introduced to help distance the game's branding from the drug in an attempt to improve the public image of the game, but also does quite well in adding to the already cryptic and mysterious aesthetic the game carries.
TL;DR: While this game isn't explicitly about drug use, the developer has said certain things that lead many to believe that the drug the game shares its name with is definitely an influence for the gameplay. Also, the game's working title was originally "Link Speed Dream" during development, but that was eventually removed as the canon title. Now the title, along with the rest of this cryptic game, is open to interpretation.