Yes Lucifer and Satan are different entities, Satan was a name branded onto so many entities.
Many have said Bael, Asmodeus and various other demons are in fact Satan.
There are two Satan’s from my perspective, there is what I would call the modern Satan.
A thought form manifested from movies, music and general media. From religious fears and the followers of the JCI funneling their views into this one astral structure. Modern satanism didn’t really help with this either, causing confusion and mixing and matching various mechanics to a machine they were creating and called it Satan.
Then we have the ancient Satan, which is the adversary. This entity is complicated one, he is the spirit of revolution and of chaos.
The ancient god Yam is I would say the original Satan. He is the god of the seas in the Canaanite pantheon. Yam is the deity of the primordial chaos, for his domain is the abyss the dark bottom of the seas. Which the Bible refers to as the abyss, the pit etc.
Working with Satan he would teach you various things like the negative and positive powers of the element water. Dark Aquarian magick, the gnosis of the serpentine path and more so.
However I would say Satan is the emissary of that one sided darkness. Which is why I haven’t worked with him all that much.
Lucifer on the other hand, is the right hand of light. The light force, the force of the celestials, the angelic, life, creation, order. Then Lucifer has the left hand of darkness. The darkness, the infernal force, the demonic, death, destruction, chaos.
Becoming the beacon of both darkness and light, creation and destruction, order and chaos, is very valuable.
So becoming powerful in you’re magical development can be done with both these beings. However my recommendation would be Lucifer.