Taking the question as written, no there are not standard rules that universally apply.
Actually determining whether someone has a medical emergency is based on assessment by a clinician, which will typically include taking a medical history, usually some physical examination, and often supplemented by other investigations to reach a diagnosis. The diagnosis will determine whether the presentation is a medical emergency or not.
Taking a step back in the process, there are many 'red flags' which if noted within the symptoms should prompt investigations for particular diagnoses. These can be as broad as 'chest pain' (which would prompt for specific further details in the history), or 'sudden onset, severe chest pain, tearing in nature, which radiates to the back' which would prompt investigations for aortic dissection.
Prior to clinical assessment, there are triage systems that look to risk assess a patient to give the likelihood that they are presenting with a medical emergency, and which can be used to prioritise patients when resources are insufficient to see everyone immediately. There are a range of these which are used in difference settings, and are not universal or international.
In the UK, ambulance services use AMPDS (also used in the USA) or Pathways. Most hospitals use the Manchester Triage System. Many others exist, some on a national basis, or for mass casualty incidents.
The risk assessment/triage tools currently rely on seeking responses to specific criteria; the information might be included within free text, but without all of the required information they could not be utilised. The actual diagnostic process needs an interaction with the patient; red flags might be elicited at an early stage within that process, but relying on a limited statement about the condition would not reliably ensure the necessary information to identify an emergency would be present.