Of course I don’t know what the situation is between you, your sister and your mom but there may be situations where your mom cannot answer for herself so her sister, or indeed yourself may feel it necessary to speak for her.
I have a similar situation with my wife. When she sees a doctor for an ailment she simply will not explain the symptoms she is suffering from and I have to step in and give an accurate description for my wife.
Another reason why you or your sister may feel the need to explain something about your mother’s condition to the doctor is is your mom has a mental issue such as dementia.
However from your question I get the impression that your mother is perfectly capable of explaining the problem she has to the doctor.
I certainly do not consider that just because your sister said her mother had sores in her throat when her mother did not may be suffering from Munchhausen by Proxy Syndrome.
Your sister may have genuinely thought her mom may have had sores but her mother did not mention it..
Munchausen syndrome by proxy is a mental health disorder in which a caregiver, most often a mother, routinely makes up fake symptoms or causes real symptoms in a child or adult victim to make it appear that the victim has a true physical or mental health issue.
In the case of your sister possibly making a mistake about sores in your mothers throat does not add up to Munchausen Syndrome by Proxy.
Added to this the only person who is qualified to make a Munchausen Syndrome by Proxy diagnosis is a qualified psychiatrist or psychologist.