It can be noted that the terminology regarding position isn't were the crown or dial points, but rather (and more logical) which part of the watch is pointed up. It's also clear that the measurement values are rounded as written.
The most common is to specify that a watch is a chronometer, which is to say it, or at least its movement, has passed the COSC testing.
COSC allows a watch to run at -4s/day to +6s/day (= +1s/day +-5s).
Grand Seiko: -3s/day to +5s/day (= +1s/day +-4s).
IWC: 0s/day to +7s/day (= +3.5s/day +-3.5s).
Bell & Ross: 0s/day to +10s/day (= +5s/day +-5s).
Sjöö & Sandström: -2s/day to +4s/day (= +1s/day +-3s).
Zeno: -5s/day to +30s/day (+35s/day for manual wind watches), if not certified chronometers.
Tissot: -10s/day to +40s/day
Patek Philippe: -2s/day to +3s/day (= +0.5s/day +-2.5s).
I have taken for granted that these accuracy promises are like water resistance and only valid for a certain time after having had a watch serviced, but as opposed to water resistance I don't know exactly how long this period is.
It's often mentioned that the COSC testing for mechanical watches (there is one for quartz watches too, allowing for a much smaller error) takes place during 15 days at five different positions and at three temperatures. This testing is heavily weighted towards the most common temperatures a watch will be subjected to (+23C unless otherwise noted) and positions a watch will be worn in (CL=crown left, CU=crown up, CD=crown down, DD=dial down, DU=dial up):
CL CL CU CU CD CD DD DD DU DU DU+8C DU DU+38C CL CL
The actual certificate looks like this.
It can be noted that the terminology regarding position
isn't were the crown or dial points, but rather (and more
logical) which part of the watch is pointed up. It's also
clear that the measurement values are rounded as written.
Note that the daily rates during testing can be outside of the acceptable range. It's the mean value, mean and maximum variation and some other values together that counts.
This article goes into more detail on what is required to pass the certification: Swiss Chronometers
The German standard DIN 8319 and the Japanese GB/T 4032-1983 are equivalent to ISO 3159:1976 which is the one COSC uses.