[Tomisimo]

What's "correct" (What they teach in classes & grammar books), and the way native speakers actually say things, are often different. This of course also depends on where the native speaker is from. In Mexican Spanish you can often use "my" and "mine" for body parts

Me duelen mis pies
Me duelen mis brazos

Me duelen los pies
Me duelen los brazos.

These are all common in Mexico, I'll venture to say that using "mis" etc is more slang and using "los" is more formal or correct.

Moving on to what your question actually gets at.

The whole thing where you should use "the" with body parts is pretty much only for reflexive verbs. Reflexive verbs are the ones ending in -se that the "action" of the verb comes back and points at the subject.

Reflexive verbs:

lavarse
dolerse
sentarse

"Me lavo las manos"
"Te lavas las manos"

"Me duele la cabeza"
"Te duele la cabeza"

"Me siento a la mesa"
"Te sientas a la mesa"

You can see with these verbs, you must use the me, te, le, se, nos in front of the verb. What this means basically is that you're already mentioning the person, so you don't have to mention it again.

Me duele el pie.

Literally, this could be "To me hurts the foot". Since you're already saying "to me", then in Spanish you already know that it's your foot, so you just say "the foot"

Now getting back to Shakira's song, "Sabes que estoy a tus pies" is not using a reflexive verb, so we don't know whose feet they are, thus you have to say "tus" (your). If Shakira was saying "I'm washing your feet" instead of "I'm at your feet" it could be "Te lavo los pies" using "los".

I hope I explained it ok, even though I rambled a lot.

Just ask if I wasn't clear in something.