These are both only partially correct.

Repeating Words

In order to get a good vocabulary score, you should try to show knowledge of a range of words, a broad vocabulary. So, if possible, use suitable synonyms and other words sometimes instead of repeating the same word all the time.

As well as that, it sometimes does not "sound" very good or, is not very good style to repeat the same key word twice in the same sentence or very close to each other, such as near the beginning of the next sentence. In this case, you can often use a pronoun such as "it", "them", "this", "that" or a phrase such as "this situation" etc. (Writing, "this phenomenon" does not impress examiners as much as you might think.)

But the idea that you should not repeat the same word more than twice is too rigid and, in fact, ridiculous in a 250 word essay. Just try not to repeat it many times, if possible, especially repeating them close together. But there are some words that don't have many synonyms or other ways to write them so, in this case, you might have to repeat the word a bit more often than with other words.

The problem with lower-level students is that they are sometimes not very sure of what are suitable synonyms or alternative ways to express a word. They then choose words that are close to but not the same in meaning, or words that change the emphasis of the original word, or words that are completely unsuitable for other reasons. This will hurt the vocabulary score, not help it, and it might confuse the reader, making the essay difficult to understand. The candidate in this case would get a better score by repeating the same word a bit more and not making unsuitable substitutions for that word.

Writing more than 300 words

It is not true that "examiners prefer more than 300 words". You get graded on the criteria that are written in the Task 2 Band Descriptor and according to how well you follow the instructions in the essay question. "More than 300 words" is not written in the Band Descriptor nor in the instructions for the essay.

Certainly, you should write more than 250 words. Some examiners might use different methods to count a word, compared to you, (such as calling "90%" one word, not two) so, to be safe, you should write at least 255 or 260 words.

It is true that, in general, you can express more ideas or points in 300 words than in 260 words and, if you successfully did that, you would probably get a better score. I say "probably", not "certainly" and I emphasize "successfully" because forcing yourself to write more than 300 words might add to the total number of errors you make in grammar and vocabulary, all for the sake of writing only a few weak extra ideas, or completely irrelevant ideas, not ideas or points that have real value in answering the essay question.

So, it is clear that writing more than 300 words does not automatically give you a better score and, in fact, trying to write more than 300 words in the real test creates several problems. The biggest problem for people at the Band 5 to 6 level is that it is difficult to write a good essay of 300 or more words in about 40 minutes. Someone at the Band 8 level could do it but a Band 5.5 candidate would be writing in a hurry, not thinking very clearly about the real or full meaning of the question, not thinking very clearly about the logic and arrangement of the ideas, and writing sentences quickly without thinking very much if the grammar and vocabulary is as good as he or she can write. And, as I mentioned above, some people who try to write more than 300 words might have already run out of relevant ideas or points and they end up writing irrelevant content (or weak ideas) that might actually damage their score, even if there are no extra grammar or vocabulary errors.

As well as that, the quality of the handwriting (the legibility) might become poor when you write in a hurry. If it is too hard for the examiner to read, he or she might not bother to, (or might not be able to) closely read what you wrote or fully consider your ideas. You need to understand that examiners are grading a lot of papers at one time and they usually can't give a Task 2 essay more time than, say, 10 to 15 minutes because of the number of papers to do. Not only that, the examiner might be getting tired after grading many papers. I think that sometimes (not always) the legibility of the essay alone might be indirectly worth 1/4 of a band point for that essay due to the effect of poor legibility, even though this is not written in the Band Descriptor.

Overall, I think candidates at the Band 4.5 to 7 level should teach themselves to write Task 2 essays that are between 260 and 290 words long. (Writing 260 to 275 words would be better.) You can teach yourself this by practicing writing essays using copies of the real exam writing paper and then you can get to know how many lines you need to write to reach the word count you want.

You should also try to teach yourself how to summarize better. In my opinion, these Task 2 essay questions really suit an essay that is 400 to 500 words long because the essays usually involve more than one point that needs to be evaluated and these points themselves are usually quite complex. When I first tried to write some model Task 2 essays, I found that my essays were in the 350 to 400 word range, which is too long. I then had to go back and cut out the less relevant content and summarize other content by rewording it. From this, I realized that the nature of these Task 2 questions is not simply to test your essay writing ability but to test your ability to write concisely, which really means expressing some things (or even most things) in a rather summarized form. This is assuming your essay has several ideas or points, not just one or two.

In the real test, you can't write a longer essay and then go back and revise it like that. So I suggest that, in your study, you first practice writing essays that are 300 to 400 words long and then revise those essays to make them 260 to 290 words. After doing that several times, you should then try to directly write in the more revised style the first time you write an essay, keeping the essay between 260 and 290 words. That way will teach you how to fit several ideas or points into a relatively short essay.

One way to avoid wasting words on irrelevant content is to not use unnecessary words, or write unnecessary content, in your introduction paragraph. Certainly, don't repeat the question, even if you rewrite the question using different words to the original wording. (You will lose some of your word count if you just repeat the question almost verbatim and if you go below 250 for your word count, you lose points.) And don't explain the meaning of the question as if the reader is an uneducated person who doesn't understand what the question means. Instead, try to write a concise and relevant piece of extra information about the topic or the question, a piece of information that is related to the remainder of your essay. (This is what "introduction" really means.) But simply writing that "this question has recently aroused a lot of controversy in society" (or words to that effect) is weak and boring, (weak because every Task 2 question is about a social question that has two or more viewpoints), is often untrue or inaccurate (for example, by writing "a lot" or "recently" when the topic may not really be discussed very much or when it has actually been a social question for a long time), and is obviously from some kind of "essay template". As a result of the non-originality of this statement (since it came from a template) and the inaccuracies, it usually fails to impress the examiner. The people who write those "templates" suggest writing about how the topic has recently become controversial for the students who lack the ability to write anything else. It's better than nothing because it does, more or less, apply to any Task 2 writing question, but try to do better than that if you hope to get 6.5 or higher.

Finally, I suggest you try to get your advice about writing IELTS essays from more than one source or more than one book, especially if that teacher or the author of the book is not a native English speaker. You should try to read the advice written by at least one native English speaker, especially the advice written in the high-quality (well-known) IELTS books written overseas. Most of those authors are not only experts in language teaching but have also been trained as IELTS examiners and have had some experience working as examiners. You certainly won't see the advice that you asked me about in those high-quality imported IELTS text books.