The verb morir is optionally pronominal, i. e. it can take the reflexive clitic pronoun even though it's not semantically reflexive. This pseudo-reflexive form has several uses, as explained elsewhere. With optional pronominal verbs sometimes the difference is subtle.

In the case of morir, the plain form is, well, plain: Murió just means "s/he died". The pronominal form usually carries a mediopassive meaning (the dying was an event that causally concerned just the person who died), and it's generally more formal and distanced, as employed e. g. when narrating a historical incident.

Thus you would say:

• Fue herido y murió en el campo de batalla. = "He was wounded and died in the battlefield."
• Murió en un accidente automovilístico. = "He died in an automobile crash."
• Murió tranquilamente mientras dormía. = "He died peacefully in his sleep."
• Mi abuelo murió hace diez años. = "My grandfather died ten years ago."

The pronominal form tends to be more expressive and it almost always shows emotional closeness. It's often accompanied by this mediopassive nuance, the idea that the dying person is dying by her/himself, or that death in itself is the process, not the end result. You don't often use the pronominal form to give explanations of fact about the person's death.

The metaphorical meaning of morir(se) por (alguien/algo) is of course always expressive, but you can use either the plain form or the pronominal one, since the above are not hard-and-fast rules. These are all valid:

• Muero por tomarme una cerveza fría.
• Me muero por un helado.
• Muero por hablarte.
• Me muero por volver a verla.

Each optional-pronominal verb has its own shades of meaning. Your second example seems to have a completive meaning, like the much more usual verbs irse, marcharse, escaparse. I've never found pronominal correrse with this meaning myself, though. My native-speaker instinct suggests that Me corría a la escuela would be rendered in English as something like "I was taking/getting myself to school", while the non-pronominal would just mean "I was running/rushing to school". There's a sense of personal finality there. Another example would be:

• Dormí mal anoche. = "I slept badly last night."
• No pude dormir. = "I couldn't sleep."
• Me dormí bastante tarde. = "I got (myself) to sleep rather late."
• No me pude dormir. = "I couldn't get myself to fall asleep."

This could go on and on; you'll get the idea, in every case, by actually employing the different verbs and paying attention to how other people employ them.

The verb morir is optionally pronominal, i. e. it can take the reflexive clitic pronoun even though it's not semantically reflexive. This pseudo-reflexive form has several uses, as explained elsewhere. With optional pronominal verbs sometimes the difference is subtle.

In the case of morir, the plain form is, well, plain: Murió just means "s/he died". The pronominal form usually carries a mediopassive meaning (the dying was an event that causally concerned just the person who died). The plain form is generally more formal and distanced, as employed e. g. when narrating a historical incident.

Thus you would say:

• Fue herido y murió en el campo de batalla. = "He was wounded and died in the battlefield."
• Murió en un accidente automovilístico. = "He died in an automobile crash."
• Murió tranquilamente mientras dormía. = "He died peacefully in his sleep."
• Mi abuelo murió hace diez años. = "My grandfather died ten years ago."

The pronominal form tends to be more expressive and it almost always shows emotional closeness. It's often accompanied by this mediopassive nuance, the idea that the dying person is dying by her/himself, or that death in itself is the process, not the end result. You don't often use the pronominal form to give explanations of fact about the person's death.

The metaphorical meaning of morir(se) por (alguien/algo) is of course always expressive, but you can use either the plain form or the pronominal one, since the above are not hard-and-fast rules. These are all valid:

• Muero por tomarme una cerveza fría.
• Me muero por un helado.
• Muero por hablarte.
• Me muero por volver a verla.

Each optional-pronominal verb has its own shades of meaning. Your second example seems to have a completive meaning, like the much more usual verbs irse, marcharse, escaparse. I've never found pronominal correrse with this meaning myself, though. My native-speaker instinct suggests that Me corría a la escuela would be rendered in English as something like "I was taking/getting myself to school", while the non-pronominal would just mean "I was running/rushing to school". There's a sense of personal finality there. Another example would be:

• Dormí mal anoche. = "I slept badly last night."
• No pude dormir. = "I couldn't sleep."
• Me dormí bastante tarde. = "I got (myself) to sleep rather late."
• No me pude dormir. = "I couldn't get myself to fall asleep."

This could go on and on; you'll get the idea, in every case, by actually employing the different verbs and paying attention to how other people employ them.