The change is mirrored in smaller towns. Wired noted that Brookline, Massachussetts recently passed a ban that exempted personal devices, while the Bay Area town of Alameda is considering similar legislation.
These loosened rules recognize a common problem: it's difficult to completely avoid facial recognition technology in the modern era. While you don't always have to use the tech when it's available, even lower-priced phones and computers may come with it as an option. Cities either have to find workarounds or treat certain kinds of facial recognition as different. Face ID, for instance, is a strictly on-device security measure where the ban is primarily targeted at surveillance.
As it is, the city's ban might have come just in time to avoid serious privacy headaches. The SFPD confirmed to Wired that it had started a 90-day test of a facial recognition-based mugshot search system in January 2019, or right when politicians were proposing a ban. The department disabled access once the trial expired, but it only "dismantled" the servers for it once the facial recognition ban took effect in July. Police officers could have been using facial recognition tech extensively without the public even knowing about it -- that alone could justify the ban for critics, even if some tweaks are necessary.