As someone who played his first mage to 60 on this server (in the glory afk leveling days and after the great pet nerf), I'll shed some light and give some rules of thumb which pet to use in what type of situation.
****Take all below as a rule of thumb and consider these are just MY personal notes. Our class has gone through a lot of changes on this server since its inception. Many of the wiki guides and forum posts were written back when pets were not classic, very overpowered, could instant cast ds, and did not eat exp. Take what others say with a grain of salt because many of people's opinions still reflect the idea that pets instant cast their abilities or even do not eat exp...
Fire / Dizil's Deafening Decoy: Lowest hp, Innate dmg shield, Has fire nuke close to dmg shield, Wizard in Kunark, DDD is essentially a beefed-up version of your 49 fire pet that procs hate instead of nuke
*Use when soloing or "pet kiting" as some call it
*Remember that the DS is NO LONGER INSTANT CAST so chaining this pet is futile UNLESS you are going to kite the mob while the fire pet takes his sweet time (can be up to 15 seconds or more!)
*Don't waste your time with your fire pet in Kunark. It's just sad. Use DDD to solo when you get it.
Water: Highest HP after Earth, Nukes, Great dps, Highest Regen, Rogue in Kunark
*A good pet to use while grouping (Insane dps in 50s)
*Kunark water pets gain the ability to backstab so position him.
*A great option when fear or aggro kiting Duo post 50
Earth: Highest HP, Root attack, Mediocre dps, Warrior in Kunark
*I liked to use when duoing with a Shaman (tanks as well as a poorly geared warrior)
*Good for spliting two mobs
*Good for chain killing
*Mediocre crowd control... but it can do it...
Air: Third highest HP, stun attack, invis when idle, high avoidance, Monk in Kunark
*Best all around pet used in groups
*Great against caster mobs (Seb comes to mind)
*Great for farming at higher levels.
Pre-50 / old world SOLO: FIRE, FIRE, FIRE...
Here is what I noticed works now. No, it is not as efficient as the "good 'ole days" but it works.
Cast your level one nuke. Send in FIRE pet. Heal him when he gets low. Reclaim him when the mob starts to flee. Finish mob with lower level nuke. Full experience. Re-cast pet. Do again. This works well once you get DDD too. Same tactics can be used to 60 albeit not necessarily the fastest. (BW giants, Scoriae Hornets, Karnor's Trash, and Ravishing Drolvargs come to mind)
Chaining with fire pets won't work anymore since they randomized the pet's ds time so unless you want to kite around the mob in order to save mana from just healing when you could be medding, I say just heal your pet up until the mob starts to flee. The fire pets are really too weak to be worth anything without their ds up.
DUO: WATER, AIR, EARTH... (All situational)
When kiting pre-50, air pet stuns are nice and water pet is always good dps. Post-50, earth pet is great when leveling with a Shaman (Slow + torpor in HS is great exp). KC entrance with a bard is an example where water pet's BS is essential whereas Chardok entrance with a necro, the stuns from the air pet will slow the mob's trek as well as save you from a mob going too far down the tunnel...
GROUP: AIR, WATER, MAYBE EARTH... (Situational)
Pre-50 I wouldn't group unless you really want to... You'll spend plenty of time grouping in Seb or HS in your late 50s. I recommend Air pet under 50, and Water / Air post 50 depending on situation (ie casters) and what is the highest pet spell you have at your current level. (MAYBE Earth for Crowd Control if some whiney group member complains enough about needing an "off tank" or "extra rooter" to which you roll your eyes and appease him until they notice the dps is worthless and our Earth pet's root is really a novelty not to be taken as seriously as the water pet's backstab or the air pets stuns.)