Both phrases you gave mean "I ate lemon". I will show you, roughly, how to parse the sentences so you can convince yourself of this.
The verb is in the end of the clause
Clauses in Japanese always end with the verb. This is very important, and (usually) won't change in casual speech, even if there are particles being omitted and all that. The verb is always in the end of the clause.
This means that, when you see:
watashi wa remon wo tabeta
You can conclude:
- This is one complete clause. The verb is "tabeta".
- What is marked with "wa"? Answer: "watashi". Therefore "watashi" is the topic of this clause.
- What is marked with "wo"? Answer: "remon". Therefore "remon" is the object.
Therefore the translation can only be "I ate lemon". Now, if the sentence was, instead:
remon wo watashi wa tabeta
The analysis would be identical to the one I did above, and the only possible translation is still "I ate lemon".
What if particles are omitted?
In casual speech, it is not uncommon for particles to be omitted. But, you can be sure that the verb will still be in the end of the clause. Consider:
watashi, remon tabeta.
Particles were omitted, this is fine. But as you can see, the verb is still in the end. This won't change in acsual speech. You simply have to guess which particle was attached to "watashi" and which particle was attached to "remon", and you can quickly guess "wa" and "wo" respectively, and understand the sentence. This still means "I ate lemon".
How to say "Lemon I ate" then?
First of all, note that "Lemon I ate" is not a clause! "I ate" is a clause. But "Lemon I ate" is well, a lemon with a special property (you ate it), so it is a noun modified by a clause.
Clauses can modify nouns. I don't know if you're familiar with this, so I will explain a bit. Clearly, we can modify a noun with adjectives, such as "big lemon", "small lemon", and so on. In english, we do this by putting the adjective on the left of the noun. We can also modify a noun with a clause. For example, "A lemon that I ate", or "A lemon I ate" for short. In Japanese, we can also do this, and to do this, we put the clause in the left of the noun. Therefore.
- We want to say "Lemon I ate".
- Step 1: figure out how to say "I ate" (the clause). Well, it is "watashi wa tabeta" or "watashi ga tabeta" (in small sentences it is hard to see a difference between "wa" and "ga", so let's leave it like this for now...)
- Step 2: figure out how to say "Lemon": that is "remon".
- Put the clause to the left of the noun. Result: "watashi wa tabeta remon" or "watashi ga tabeta remon".
- Note: It's unfortunate that we stumbled into this "wa" versus "ga" thing here, but since it's not the focus of your question, I will simply say that "wa" is wrong in this case. If you don't understand why, you'll have to look around or ask another question.
Conclusion: "Lemon that I ate" is "watashi ga tabeta remon". Since clauses in Japanese end with a verb, it is clear that "remon" is not a part of the clause here. Instead, it is being modified by a clause. And no particle omitting will change this.