Just to update based on experience and the latest edition of IEC 60601-1-8.
The 2003 edition (including A1:2006) still does not have a definition of an "alarm". So it remains an open to interpretation and semantics such as "low battery alarm" vs "low battery alert".
The 2006 edition (which fits with the 3rd edition) is also missing a definition. However, it does in effect have a definition via Clause 4. That says that if a risk control involves providing a notification to the operator, that is an alarm.
That would appear to include a wide range of notifications including a low battery indication (whether or not it is called an alarm).
However ... it turns out I misread the standard before. If you read it closely, most of the requirements are preceded by an "if ..." , such as the 4m-away indication and audible alarms. The minimum, mandatory technical requirement for a low priority alarm seems to be a visual indication of the type of alarm, such as an LED next to a "low battery" symbol. This indication needs to be visible from 1m or the operator's position under both bright and dark conditions (100 - 1500 lux). It is also possible via Table 1 to judge that "no alarm signal" is required, in which case, no technical requirements apply to the means of notification.
So, you end up with a standard that is applicable, but most of the content is not applicable. Since many devices will have some kind of indicator which fits the definition in Clause 4, it would make sense if in the future that the requirements are covered by the general standard, by including a visibility test for indicators. Currently this is missing from the general standard, only markings are checked for visibility. So if you had a indicator with a weak LED that is not easy to see in 1500 lux environment, you need to reach for IEC 60601-1-8 to fail the design.
If this test were shifted to the general standard, the alarm standard could then be reserved for alarm systems that contain at least one medium or high priority alarm, or a low priority alarm that generates an audible signal.