We gotta be careful here about mixing up the terminology.
This tensor wiki page points out that a tensor of rank 3 might have different dimensions
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For example, a rank 3 tensor might have dimensions 2, 5, and 7. Here, the indices range from «1, 1, 1» through «2, 5, 7»; thus the tensor would have one value at «1, 1, 1», another at «1, 1, 2», and so on for a total of 70 values.
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In that terminology, a vector would have rank 1, but n dimensions. Saying "a vector is a one-dimensional array" is the same thing as saying "a tensor (vector) has rank 1". I can see the possibility for confusion.