X-Men: First Class has the coin used by Sebastian Shaw which later is used by Magneto to kill him, himself succeeding the villain he fought against. Cerebro doesnt appear, because it was constructed as a prototype in the film.
Dark Phoenix, while being the most toned-down title sequence (no Cerebro), features fire in the film title. The Phoenix is shown to be both villainous and heroic in the film as well as not being the main villain of the film (that's the D'Bari), so a traditional title would not have worked out.
The connection between The Gifted and Legion and the X-Men films can be found if you take into account a recent episode of Legion where David talks about the theory that their are strings of alternate timelines that diveverge and diverge at different points due to different decisions.
Past Xavier (McAvoy) seems to be a lot more cocky and overconfident since X Men Daysof Future Past. Well, obviously: hes changed history, defeated a mutant god and mentally controlled the mutant who defeated the aforementioned mutant. With that said, his overconfidence does get the better of him on those occasions.
Future Xavier (Stewart) never went through those events. So he appears more sober and serious, and doesnt underrate anything or anyone. His battle with the Phoenix in X-Men: The Last Stand was for the first time and he came off the worst for it, because he never had previous experience with it or anything like it.
In contrast to the above, Past Erik (Fassbender) becomes markedly less daring. He doesnt have a Breakthe Haughty arc, but he is more cautious and willing to keep his head down and look after his own, only rising up to retaliate under situations of major loss (his familys slaughter in X-Men: Apocalypse and Ravens death in Dark Phoenix).
Future Erik (Mc Kellen) is more ambitious and willing to endure risks and losses (including mutant lives, even those of close ones) to fulfill his goals. He went too far when he sought to manipulate the Phoenix, which went on a destructive rampage that shocked even him; his past version in contrast sought to have the Phoenix eliminated.