Writing Down Chinese Names In Correct Order
Chinese names are written (in Chinese) in the following order:
surname given name
In non-Chinese contexts, such as a database of mixed-nationality names, all names (Chinese or not) should follow the order given by the target language.
Taking one of the examples, Lu Xiaojun vs. Xiaojun Lu, the Chinese name order is Lu Xiaojun (surname, then given name), while an English name order would be Xiaojun Lu.

Note that, for three-syllable names like Lu Xiaojun, it is often easier to determine the words corresponding to the given name and surname. Two-syllable names (e.g. Li Ping vs Ping Li) are more difficult; we can make educated guesses (the surname in this case is probably more likely to be Li), but Chinese characters are necessary to confirm this.
If you expect that at least +90% of the names are in the correct order, you can make an inference for classifying the surname and given name words for the two-syllable names, based on the existing three-syllable names, but this is really just guesswork.