Is the sentence "I want to take a rest" wrong?

Grammatically speaking, both usages are fine, as @Matt points out. But idiomatically it's a completely different story. These figures from Google Books reflect my own gut feel...

1: I want to take a rest - 7 results
2: I want to take a break - 462 results
3: I want to rest - 707 results

The disparity in usage between rest/break in the first two is specific to the verb take, as can be shown by the fact that have a rest and have a break both return about 1500 hits.

I don't think any native speaker would consider the first form "evidence of a non-native speaker", but it certainly seems fair to say that whoever advised OP against using it has some justification.

Apart from this idiomatic preference issue, I think any attempt to apply semantic distinctions is largely pointless. All versions effectively mean the same thing, and are equal in "formality".

Both are fine. "I want to take a rest" sounds a little more informal, and probably would sound a little more natural in spoken conversation.

Both generally mean the same thing.

However, since you're looking for a way to convey the message as a native speaker, I would recommend you drop both terms and instead use: I want to get some rest.

I want to take a rest

You might say I want to take a rest when you want to rest from doing something.

For instance,

I've been moving furniture all day. I want to take a rest.

Hold on, I just finished a long run, I want to take a rest.

Although, in both cases, it may be more appropriate to say I want to take a break or I want to get some rest.

I want to rest

The term I want to rest is less common.

Saying I want to rest before dinner may sound better than I want to take a rest before dinner, but in either case, I want to get some rest before dinner is more likely to be the preferred choice for a native speaker.