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In the context of computers there are containers, e.g. arrays that assign indexes to their elements beginning with 0 for the first element 1 for the second element, 2 for the third element, etc.

There are also containers that start counting with 1 for the first element and 2 for the second element.

Is there a single word that denotes this property?

Example:

The coordinate array is ___ , you can access the first element with the index 0. The polygon set is ____, the first element has the index 1.

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The words you are looking for are zero-based and one-based:

Zero-based numbering or index origin = 0 is a way of numbering in which the initial element of a sequence is assigned the index 0, rather than the index 1 as is typical in everyday non-mathematical/non-programming circumstances.
Wikipedia: Zero-based numbering

1-based indexing
Definition: Indexing (an array) beginning with 1.
Paul E. Black, "1-based indexing", in Dictionary of Algorithms and Data Structures

For example:

I've also heard of zero-indexed being used, but I guess it's less common.

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    The main use of zero-indexed (IMO) is to avoid using the same word as in base 2, base 10 etc.
    – Chris H
    Jan 22, 2018 at 17:36

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