Is the daughter considered sinful for bearing the guilt of her mother's severance and anger, because she asked for simple financial help to perform Umrah, while the mother helps her sons with larger amounts?

You did not err in requesting the loan from your mother. However, if you rebuked her in an inappropriate manner, then seek Allah's forgiveness and be gentle in reconciling with her. You must mend the relationship between you by striving to speak with her, visit her, kiss her head, and apologize to her until she is pleased. Seek assistance from someone who can mediate the reconciliation. Seek reward and supplicate to Allah. If she is still not pleased after you have exerted your utmost effort, then there is no sin upon you, and she bears the sin of severance.