Partners and children are covered irrespective of being an additional card member (see the definition of "primary card member" on pg 9).
We are talking about two different things here.
a) Partners and Children are covered under the definition of Primary Card holder.
b) Additional Card Holders are not covered.
If Partners and Children are also Additional Card holders they are covered, but only under (a), not (b) when travelling with the Primary Card holder.
The OP's warning talks about Additional Card holders not being covered when not travelling with the Primary Card holder.
Additional Card holders are not covered at all... period.
If the Partner is a Primary Card holder they are covered under (a).
If the Partner is a not a Primary Card holder they are covered under (a) if travelling with the Primary Card holder.
If the Partner is a not a Primary Card holder they are
not covered under (a) if travelling without the Primary Card holder.
Nobody is covered under (b)