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
Incarnate Construct—the only true negative, −2, but still +0 total
The incarnate construct template from Savage Species turns a construct into a humanoid or giant, and reduces the LA of the construct by 2. It cannot reduce a character’s overall LA below +0, so for a construct that has less than LA +2, other templates must be applied to get the full benefit. Like dragonborn, the template is very cheesy with warforged.
Xvart—LA +0, but at some point a fan list typo’d it as −2
Xvarts from Dragon vol. 339 are sometimes claimed to have negative LA—they do not. They have LA +0, and a negative CR adjustment. There appears to have been a fan listing that put the CR adjustment as the LA and this gets repeated sometimes.
Epic Level Handbook—unadulterated garbage
Epic Level Handbook is a horrible mess, and basically unusable for any purpose, but it’s particularly poor in this case: its description of level adjustment is completely wrong and makes no sense. It suggests that ECL is LA + class level only, and then seems to kinda-sorta imply that PCs don’t actually get racial hit dice at all?
None of the monsters in Epic Level Handbook even have an LA assigned to them—not even the winterwight discussed as an example in the level adjustment section, or the worm that walks template which discusses how to make characters with it. The 3.5e update PDF for Epic Level Handbook gives an LA +7 to the mercane, but that is literally the only one.
But nonetheless, on page 156 there is a table “Monsters by Challenge Rating” which also includes an “ECL” column which indicates a number for several creatures. The number for the winterwight, for example, is 25—which matches the LA +25 that the example on page 155 used. And that is despite the fact that the winterwight has 32 HD. That means that you are, I guess, expected to deduct the 32 HD and their associated benefits from the winterwight, apply what’s left to your character along with their 1st class level, and call that a 26th-level character. It’s not, in case you were wondering.
But the real problem here is that the Epic Level Handbook doesn’t get to redefine ECL and LA—per the errata rules, this contradiction should be ruled in favor of the primary source, which is going to be the core rulebook in the case of a contradiction between a core source and a non-core source. So despite what Epic Level Handbook says, PCs do get the RHD of these creatures, and LA is then added on top of that. Untangling this mess becomes impossible. You could treat the “ECL” column of that table as in fact being the “LA” column—after all, Epic Level Handbook claimed they were the same—or you could treat it as actually the total ECL, and back-calculate an LA value for each creature—which will in many cases be negative.
Or, you could do the best thing, and throw the Epic Level Handbook in a fire. It will serve you far better providing warmth and light than it will providing you D&D 3.5e rules.
Negative character level adjustment doesn’t exist
There is definitely not any way to have a character with negative LA. If there was, it should be banned at every table.