I have 5 lights and 2 switches. When I press switch 1 I want light numbers 1 and 2 and 3 to come on and when I press switch 2 I want light numbers 3 and 4 and 5 to come on. Note that the 2 switches are on two different circuits in the distribution-board but are on the same phase.
You'll need two double-pole switches. Those are front and center in my illustraton. There are no relays.
Since you are using separate circuits you must maintain separate neutrals and keep them rigidly separated. You cannot cluster the neutrals. (Grounds you can, neutral is not ground). This applies even if they are the same phase.
In the U.S., gray is an allowable neutral color and this is why: so you can distinguish differently sourced neutrals in close proximity. I am using purple for the "hot" associated with the gray neutral, and red for the other hot. This is for illustration; if you are in conduit you can use those colors straight up, otherwise you will be stuck with black and white as found in Romex. You can use tape to mark conductors as different, though, best to use gray tape since black tape on a neutral is usually assumed to be a hot. You could also tape the wire bundle.
The center box is your junction box with the 2 switches. The bottom box is your circuit breaker panel.