OH, It is quite a lot more than that.

Two different things--
Loving-kindness (as understood by Theravadins) is the wish that other be happy.
Compassion (as understood by Mahayanists) is the wish that other be free from suffering.

Bodhicitta which is often translated as "Compassion" means Awakening-Mind.
There are two kinds of Bodhicitta, Relative and Ultimate. The Relative, is the
Altruistic Intent followed up with actions; the Ultimate is the direct Recognition of Emptiness.

As a practice, Relative produces Ultimate. The Path is outlined in core texts such as
the Bodhicaryavatara by Shantideva and invloves the six perfections, the five paths,
the four immeasurables (Bramha Viharas).

Shantideva said it most concisely and beautifully:

"Whatever joy there is in this world
All comes from desiring others to be happy,
And whatever suffering there is in this world,
All comes from desiring myself to be happy.

But what need is there to say much more?
The childish work for their own benefit,
The Buddhas work for the benefit of others.
Just look at the difference between them!"

"Working for the benefit of others" begins with generosity.

What you explained Laith, is more like the western psychological view of 'empathy.'

Compassion in Tibetan Buddhist terms, while not necessariliy void of such a feeling as you mentioned- is defined as the experience of wanting to remove even the slightest suffering and the causes of that suffering from the experience of all samsaric beings. In addition, it is free of sentimental attachment, a notion often confused or misrepresented as compassion and/or loving kindness.