Im no electrician but ive some personal experience... disclaimer
Its irrelevant with modern electronic timers (as opposed to the "mechanical" magnetic ones that warning is attached to), they have contactors rated for 10+ amps usually (another important thing to check when buying a timer) and will not fuse together.
Also modern digital ballasts are all soft starting (again as opposed to old magnetic impulse ones that were not to be used with plug and play time relays) and have built in circuit protection.
Ive had both types of timers and ballasts. Got rid of the impulse starters soon as i saw how dangerous they are to leave unattended (coil temp of 100c). The one and only time i used a mechanical timer its contactor was fused.
Scary shit for a grow at a forest cabin yes, but imagine if its your apartment...
So, only buy digital ballasts, only use electronic weekly timers with relay protection rated for 10+ amps.
The plug and play ones usually come in 10 or 16A varieties.