Even though it isn't listed on the order form, the Ron Cey button certain looks similar in design
to the Reggie Jackson button that is pictured on the wrapper:
Based on the names on the checklist, I decided to see if I could find some other buttons, and guess what I found. Some of the buttons have the exact same picture as the player's 1981 Fleer card:
(Note the Garvey photo appears to be taken in the same location as the Ron Cey photo and in a similar pose kneeling with a bat)
Other pictures were not exact matches, but were certainly taken at the very same time:
I also found some pictures of proof sheets, which also feature similar shots to the 1981 Fleer cards:
From the proof sheets, we can see the name of the company that was producing the buttons - Sports Photo Association.
I'm guessing that since in 1980 Fleer would have needed pictures for their inaugural set that they were putting together for 1981, it would have made sense for them to work with Sports Photo Association to get images for their cards. That would explain why so many of the pictures on the buttons either match, or are almost identical to the 1981 Fleer set.
It would also help explain why there was an ad for the buttons on Fleer's wrappers. Perhaps as part of the arrangement for providing photos, Sports Photo Association wanted a way to promote their buttons and Fleer agreed to include an ad on the wrapper.
As the Ron Cey button shows, not all buttons feature photos that match or are similar to the 1981 Fleer set. For example, here is the Eddie Murray button
In looking for pictures of examples, I sometimes saw these referred to as being from 1978 and other times from 1981. It could be that SPA also made buttons back in 1978, but with so many of the pictures matching up with 1981 Fleer cards, I would have to say that their biggest set was the 1981 release.