If you can find the original pension papers, or the pension index card, they may reveal more clues.
The 1888 one may be the veteran himself, or could be an app by his widow. The name of the applicant (soldier or widow, or even surviving child by a guardian) will be in the index card, along with the state the applicant lived in. The detail file will probably have the actual address the checks were going to be mailed to.
The 1912 one is most likely a filing by the widow shortly after the veteran's death. This is just a guess based on probability - 50 years after the war. Again, the detail records will give you more, including possibly the date of death of the veteran. If you already know the date of death of one of your two vets, you can associate the 1912 pension app to a 1910-1911 death as very likely.