I see what you mean, the discrepancy is either a transcription error to the MIC or could relate to the fact he has no known grave; as both the SDGW and CWGC databases show the July date, that is more likely to be the official date of his death.
Without any service record and/or a copy of his death certificate I can only speculate that the February date was when he was first reported missing and the July date was when he was confirmed killed.
The official date of death will be as per his death certificate that can be obtained from the GRO at Southport see
http://www.gro.gov.uk/gro/content/certificates/default.asp or click on “Contact us” in RH menu box for telephone and other ordering methods.
Full reference:
GRO War Deaths Army Other Ranks (1914 to 1921)
Name: Chadwick, Alexander N; Rank: Pte; Number G/6067; Unit; R.W.S.R.; Year: 1917; Vol: I.67; Page 89.
As I was preparing this Rob posted his response.