No, the Siren does not receive a +1/+1 counter. Although your understanding of the word 'Target' makes sense in english, it has a rather specific meaning in Magic.
A creature is only targeted by spells or abilities that actually have the word 'target' on it. For instance, Giant Growth. From the Comprehensive rules:
114.1a An instant or sorcery spell is targeted if its spell ability identifies something it will affect by using the phrase “target [something],”
The Siren needs to be identified as that 'target [something]' in order for Heroic to trigger. On instants and sorceries, this is spelled out; casting an Aura (ie Unholy Strength) on War-Wing Siren also triggers the Heroic ability, as they have an implicit target:
114.1b Aura spells are always targeted.
Casting an equipment would not trigger the ability, as they enter play unattached and not targeting anything. Their equip-ability is not the same as 'casting a spell', so it doesn't trigger Heroic. The same goes for other activated abilities with a target (Spikeshot Goblin does not trigger Heroic, also see this question).
To look a little closer at what spells do and do not target the War-Wing Siren:
- Twiddle would trigger the Heroic ability.
- Turnabout wouldn't, because the card doesn't target a creature, it targets a player.
- Call to Glory would not trigger the Siren's Heroic ability, as it doesn't target anything.
(Thanks to @diego for suggesting these cards)