Alternative definitions
There is no unique definition of quartiles or percentiles.
We introduced the idea of percentiles by reading them from the cumulative distribution function. However this is a step function and computer software effectively uses a smoothed version when it reports percentiles.
In practical terms, the difference is unimportant.
If your conclusion about the data would change with a different definition of the percentiles, you are over-interpreting the data.
Annual rainfall in Dodoma, Tanzania