How much do letters of recommendation help in a statistics PhD program application?
I have no direct experience with statistics programs, but I know of no reason to think they differ from math programs in this respect.  Assuming that's the case, letters of recommendation are absolutely critical, at least in U.S. universities.  I agree with the person who told you they practically are your application, and I'm mystified by how someone could tell you they make no difference.  (I wonder whether it's someone who has never served on an admissions committee or was educated in a very different system.)

It's true that just about everyone gets what might appear to be good letters.  The issue isn't whether they say bad things about you, but rather how strong they are and how compelling a case they make.  I've certainly seen many cases of letters written by the same person that differ enormously in their effect, even though they are all nominally positive.

The third piece of advice you received in no way contradicts the second.  It's not enough for a letter just to include flattering comments about you.  It needs to make a compelling case for why you will be successful in statistics, written by someone who knows exactly what it takes to succeed in this field, has seen other strong students to whom you can be compared, and has a reputation to protect that will keep them from exaggerating or trying to manipulate the admissions committee.  A senior statistician is the best case, but a junior statistician or a mathematician may also be able to do a fine job.  A letter from a chemist will not be helpful, and a letter from a professor of literature, no matter how enthusiastic it is, will hurt your case (by demonstrating that you have no idea how the application process works or that you couldn't find anyone more relevant who thinks highly of you).

Exactly what is required depends on where you are applying.  At the top departments, you need letters that make a very strong case indeed.  Even outstanding students will be rejected, and you need letters that set you apart from the others.  At much lower ranked departments, it might be possible to make a favorable impression based largely on grades and test scores, but even then you'll still need good enough letters.  (The bar will just be lower.)