It is confusing. The septum pellucidium is a thin membrane separating the left and right lateral ventricles. It contains white matter (nerve fibers), blood vessels and a few neurons. It has traditionally been thought of as just a separating membrane, but the fibers running through it connect to the hippocampus and hypothalamus and so it is probably a relay to those structures.
Lateral to the septum pellucidium are the septal nuclei. The septal nuclei are NOT considered part of the septum pellucidium (or vice versa). The septal nuclei are traditionally divided into two major subregions, the medial and lateral septal nuclei. As you might have guessed, the medial nuclei are immediately adjacent to the septum pellucidium. As the septum pellucidium is quite thin, the right and left medial nuclei appear to compose a single midline nucleus with the septum pellucidium running through it. The lateral nuclei are the ones a bit further out and it is these that are called the lateral septum. The septal nuclei may be further divided into caudal and ventral regions. Many neuroanatomists group the most posterior parts of the septal nuclei into a third group, the posterior septal nuclei distinct from the lateral and medial divisions.
This is part of the problem. Structures in the brain were named based of where they were and what they looked like since their functions were unknown. For example the hippocampus was given this name because it looked like the tail of a seahorse (genus Hippocampus), and the locus ceruleus ('blue place') looks blue. As we have learned more, some neuroanatomists have divided regions into subregions or sometimes lumped regions together based on their models of the proposed physiologic functions or neurochemistry of those regions. The subcortical 'limbic' nuclei, including the septal nuclei are among those most subject to these changing definitions,. And as you might guess, different neuroscientists often disagree about these divisions. Being human, they sometimes behave as though their definitions are THE definitions, or gloss over differences between sometimes unclear definitions. Ideally, people splitting nuclei into proposed subregions try to propose clear definitions, but this is not always the case and these proposed definitions may not be accepted or used. This can make it difficult for experts to know exactly what regions are being discussed, let alone people not long immersed in the field. Over time however, fields tend to come to a consensus on terminology though for many brain regions this consensus remains elusive. I hope this discussion makes the topic of the lateral septum a little more comprehensible, if only by validating your sense that the topic is confusing.
Here is a readable overview about the septal nuclei and septum pellucidium:https://www.neuroscientificallychallenged.com/blog/know-your-brain-septum
Check out the first reference here to discussing the neurochemistry of the medial septal nuclei from a chapter by Risold in the Rat Nervous system https://www.sciencedirect.com/topics/neuroscience/medial-septal-nucleus
In contrast look at the introduction of the following paper regarding the lateral septal nuclei:
"Whole-brain mapping of projection from mouse lateral septal nucleus"
And if you have access, you may be interested in
"Differential and complementary roles of medial and lateral septum in the orchestration of limbic oscillations and signal integration"
And here is a paper that describes development of the posterior nuclei as being different from that of the lateral and medial nuclei