I'm sorry to hear about the challenges that you're dealing with regarding your mother. However, several questions come to mind. Is this a relatively new behavior for your mom, or
has she previously been diagnosed with some form of dementia? Has your mom been evaluated by a neurologist? Has she had a physical health assessment done to determine what it is she does or does not have?
Subsequent to a complete neurological and physical assessment, hopefully you can be told what it is your dealing with in this case.
It's very possible that your mom's behavior, which is not uncommon, is due to age-related dementia, which may or may not be Alzheimer's disease. Some elderly patients have the ability to turn their emotions and behaviors on and off to suite a particular situation. That is why your mom's visitors may not be seeing the same behaviors that you and the respite care provider are seeing.
While it may seem hard to believe, it's possible that the conversations that involve your mother and her visitors have been somewhat consistent enough in their content, that mom has actually developed responses to inquiries such as, "How are you feeling?" or "You look wonderful" that give the appearance that she's 100% on the ball. It's also possible that your mom is able to act "normally" for short periods of time when she feels it is appropriate to do so. But when the visits end, she quickly falls back into what you see as her more usual emotional behavior patterns.
As I stated, I don't know what it is that is driving your mother's behavior, but I can tell you that you're dealing with someone who in her own way is telling us that she needs a medical evaluation. I believe that your questions will be answered once you have had mom thoroughly examined.
Be sure that you document the odd behaviors and severe mood swings so you can explain her abrupt behavioral changes to the neurologist. Keep in mind that mom may try to convince the doctor otherwise by "being on her game" if in fact she is able to turn her behavior on and off. However, a good geriatric neurologist, armed with the observations you documented, should be able to provide a clear assessment of what's going on with your mother's behavior patterns.
The result of the neurological assessment may be a prescription to help your mom maintain more consistent moods, a referral to a psychologist for an evaluation and treatment for depression, or perhaps lead to the discovery of a physical problem such as a urinary tract infection, that once treated, clears up some of her behavior issues. I don't know which way this will turn out, but I do believe this is where you should start.
Best of luck, as it seems your about to embark on an interesting search for the solution to a challenging mystery.