In a nutshell, Laguna I has a key rather than a card, the bonnet sits on top of the wings, and the door handles are flush with the car body. The phase 1 L1 has elongated front fog lights (or blanking plates if fog lights aren't fitted) and glass lenses on the headlamps. The phase 2 has round fog lights (or blanking plates) and plastic covers on the headlamps. On the rear of the hatch, the phase 1 has smoked reverse and indicator lamps, whereas the phase 2 has shiny, clear ones, but the tail lights weren't changed on the estates.
The Laguna II has the card rather than the key. The bonnet sits between the front wings. Door handles protrude from the doors. The phase 1 has a single piece radiator grille with the Renault badge on the bonnet, and a large matt metallic effect trim between the bonnet and the grille. The phase 2 has a two part radiator grille with the badge between the two sections of grille, and a thin chrome 'moustache' running under the headlights and grilles. Again, the tail lights are different on the hatch, but remain unchanged on the tourer.
An 03 plate will be an L2 phase 1.
In a nutshell, the 'phase' number increments when they radically change the styling without actually changing the body shape of the car underneath. Usually this means new lights and bumpers, but there may be other differences. The 'model' number increments when they replace the main body shell of the car with a new design, at which point the 'phase' number reverts back to 1.
Edit: Here's a pic of both my old and new Lagunas. phase 1 on the left, Phase 2 on the right
Time-line wise, I think the original L1 phase 1 came out on an L plate, which would make it about 1994. The phase 2 came out in 1998 - I had one of the first of them on an S plate. I think the L2 phase 1 is generally accepted as coming out in 2001, though while I've generally only seen them from Y reg onwards, I did once see an X reg one that didn't appear to be a personalised plate. The L2 phase 2 was released in 2005.