You're talking about something very different called a hot start. The GP is discussing time to fix in a cold start scenario. You'd only see this on a phone that had been powered off for months, or "teleported" hundreds of miles away. In this scenario the receiver has to download the new time, new ephemeris data, and a new almanac (up to 12m30s in the worst case) before it can fix. Depending on the receiver, there may also be a delay of several minutes before it enters cold start mode.

If the receiver has recently (last few days) gotten a fix and hasn't moved too much from that fix, it'll be in at least warm start mode. It still needs to download ephemeris data, but this usually takes 30ish seconds to fix.

If the receiver has seen a fix very recently (last few hours) or a recent network connection, it can fix from hot start like you saw, which only takes a few seconds and may not even be observably slow depending on how the system is implemented. Phones go to great lengths to minimize the apparent latency.