You have the elemental associations correct. Some traditions though switch them for the athame and wand (athame-fire and wand-air). How you view that is a personal matter.
Another thing linked to tradition is what tools you use and keep on your altar. Some witches include a bell, incense burner, cauldron, deity statues, deity candles (instead of statues), bowl of salt, oil burner, and more. Other tools that don't quite fit on the altar that some use are staffs, swords, besoms (broom) and special robes to wear during ritual. Each person decides what they prefer the most and go by that. So unless you follow a set tradition and go by its altar set up that tells you what tools you should have, no two altars will be the same.