Level Adjustment does not convey any hit dice alone, it simply adds to the effective level of a character.
Effectively, your first option is correct in that this character will be 2 class levels behind other party members.
For example a level 1 Drow Rogue would only have a single hit die and only the class abilities of a level 1 rogue, but would count as a level 3 character and thus would need a total of 6000 xp (the amount to reach level 4) to become a level 2 rogue.
Note that some "monsterous" races may convey racial hit dice as well as possess a level adjustment. For instance a Gnoll provides 2 Racial humanoid hit dice as well as a +1 LA. Thus a level 1 Gnoll Rogue would have the abilities of a level 1 rogue, 1 rogue hit die, 2 humanoid hit dice, and the +1 LA. This character would count as level 4, but having 3 hit dice would have 2 feats (one for level 1, one for level 3).
I'll restate your question with my words to see if I've understood it properly first.
Let's keep the math easy. ECL 2, starting level 2 (1000 XP)
You get a +2 LA somewhere during level 2, let's say right after leveling up.
So you're now 1000 XP, ECL 4 and you need to reach ECL 5 for adding your new class level.
Your DM says ECL 5 is reached at 10k XP so you now need 9k XP to level up.
You say LA +2 is like going from 4 to 5 so you need 4k XP to level up.
Your way, you gained two levels (your LA) for free.
DM's math: 0 + 1000 + 2000 + 3000 + 4000 = 10k
Your math: 0 + 1000 + ............. 4000 = 5k
Your DM is right and no, it's not like you need more XP the higher you get, because your class levels you already took costed you less than if you took them after the template, so the thing balances out. The only real imbalance comes from the fact that you get the template before paying the XP for that LA.
As for the example you quoted, I'm pretty sure that was a typo and that's 9000 XP, not 3000
Best Answer
Unearthed Arcana is a little vague on this subject, but there isn’t much support for allowing that based on what we do have.
Note that the rules don’t actually call out how or why you have take three “additional” class levels. It just says “use the creature’s current level adjustment to determine the point at which the level adjustment can go down by 1.” Taken literally, that would imply that you need 3 class levels—which this drow already has by virtue of being 6th level—to get rid of the remaining LA. But the textual example, as well as the accompanying table, clearly indicate that you cannot count the 6 levels used to reduce the LA +2 to LA +1 towards the class levels needed to reduce the LA +1 to LA +0. The LA +1 can only be reduced at 9th.
There are two possible interpretations of this: you need to ignore class levels already used for other LA, or buying an LA off is a process that takes 3×LA levels. The distinction here could matter: if you got the LA +3 template at, say, 12th level, it would be unclear if levels 10th, 11th, and 12th could be applied towards buying it off. Even though those levels weren’t used to buy anything off, you didn’t have the template yet to start buying them off. You could conceivably have to start on your six levels at 12th.
The other issue is how separate instances of LA work together, or don’t. Unearthed Arcana doesn’t address this at all (and, in fact, the authors seem to only be considering racial LA, and the variant is found in the Races section of the book). Is a separate LA +3 template treated as a new, separate thing, or is it treated as adding on to your total “pre-buyoff” LA? Certainly, if you started with an LA +2 template and an LA +3 template, you would have LA +5—Unearthed Arcana doesn’t give any indication that you should be able to buy that off more than once, at 15th. But then, like I said, Unearthed Arcana seems to ignore the possibility of multiple sources of LA altogether.
So really, any one of these possibilities is true, depending on how you answer:
Where Acq is the level at which you acquired the LA +3 template after buying off an LA +2.
But there really isn’t much evidence for allowing you to start immediately at 9th: the example of the drow, who has LA +1 after buying off at 6th, explicitly says that despite having three class levels, they need three additional class levels. That is going to apply to you too.
In reality, I strongly urge you and your DM to just eliminate LA from your game altogether. Don’t play things with LA, don’t allow things with LA. The system just doesn’t work. It’s a shame but it’s what it is. Homebrew a non-LA way to get things you want/are appropriate to the character, whether that’s a scaled-down version of a creature or a prestige class that gives the template as a reward.