If you are both regular POA and healthcare POA, does that mean you have to also be the primary caregiver?

If your father has dementia he can no longer change the POA. This must be done by a competent person.
Wills and Trusts have nothing to do with POA whatsoever. A will appoints an executor to manage and distribute the estate AFTER DEATH. A Trust appoints a successor trustee to manage the trust and/or distribute it AFTER DEATH.
A POA manages a person who is incompetent to make his or her own decisions, yet the appointment of the POA must be done when that person IS COMPETENT.
I feel much as your Sister does. I would not be doing all that work and having the senior in my town without being the POA. The POA is the one who does that work, so that means you could take a vacation where your senior is going to move and get him all moved in, make yourself responsible to pay his assets for his care, have yourself registered with the banks as financial POA there, have yourself registered with MDs as medical and general POA where he will be living.
Why would you remain the POA if you are not doing the work? Why would your sister want to take on this onerous task if she is not the appointed POA, and as I said that can no longer be changed.
You CAN, as POA, appoint someone to do jobs, tasks, and pay them to do so. Say a Fiduciary. Or you could hire your sister and pay her out of father's assets if that can be afforded, or, as I said, take off some time from your job to do this. Or you can hire a Licensed Certified Fiduciary to do these tasks; that is within your power as POA dependent on how the POA is worded. A POA document tells you exactly what you can do.
I sure wish you the best. This is a big job. Sis is appointed to do it if you resign, if you can no longer manage it, or you wish to resign and have her automatically take over.
Best of luck.

No. You do not have to give over your POA for your father to be moved to a different facility in your sister's town. The terms of the trust do not change because if you give over POA. The POA is not responsible to literally do the caregiving for the person that have it for. They are responsible to arrange care and make sure the person they are POA for is being adequately cared for. Also, to make sure their funds are being administered properly.
Am I correct in assuming that your father made an Irrevocable Trust? Meaning that the Medicaid look-back period has passed and now his assets are Medicaid-exempt. Changing the POA does not change the conditons of his trust.
As for his will, whoever is named as the executor in that document is the one who is in charge of administering the will after his death. POA expires when someone passes. Their will is handled by whoever is the executor of their will.
It would make sense for you to transfer the POA over to your sister if you are going to be living 3 hours away from where he is. Your sister will be the one handling the day to day business for your father and you should transfer it over. The terms of his trust and will won't be affected by it.

Think it makes perfect sense if your dad close to your sister and you asked her to look for care where she lives … she should be primary POA

No, POA does not mean that you do the Caregiving. But I am with that if Dad is going to be 3 hours away near sister, then your sister needs to have both POAs. Read your POAs to see how being a successor works. Do you have to die before sister can take over, can you just revoke your POAs and then sister can step in. I get from Your profile you really don't want the responsibility. If Dad is still competent, have him do new POAs making sister primary and you secondary.

You don't need to be POA just because your the oldest or nearest to Dad. I really think the person who is assigned POA should sign saying they accept the responsibility. There are people who have no idea they have been assigned.

No. Nothing hasto change as far as POA is concerned in order for your father to be movied into assisted living. It would be a good idea for you to transfer primary POA over to your sister if your ather is going to be moved to her town and she is going to be responsible for all of his day-to-day decisions. It makes sense to transfer it to your sister.

POA is not the same as caregiver. POA gives that person responsibility and power to make decisions on behalf of the other person. Will does not have to change either. Nor does the trust.

I can understand your sibling's desire to obtain the POA. If she will be "the closest person," she expects that she will be called on for decisions and during medical emergencies. This does not need to happen. You can be the primary POA no matter how near or far you live from your loved one. I suggest you talk with her some more about what she is concerned about if she does not have POA.

The answer to the questions in the last 2 sentences of your post is ‘no’. Nothing ‘needs to be changed’ and Primary POA is not ‘responsible for being the caregiver’. In fact quite possibly nothing CAN be changed because of a lack of legal competence. That’s all easy.

The difficult bit is WHY sister said “if he goes to her city she needs to be Primary POA”. Has she been told this, and if so by whom? Why does she think it? And lastly, is it a good idea anyway? We do hear stories of a POA who refuses to authorise or allow payment for things that the actual carer thinks are important (about money – maximise the POA's inheritance). We also hear stories of a POA who benefits themselves and can’t be challenged (also about money).

So it’s a good idea to find out who is pushing this, to assess your mutual ‘trust’ levels, and to talk through what happens if there is a disagreement.

No, not at all. Your sister can be your dad's daughter without having power of attorney. She's wrong.

The one with POA is in control of the money and property. If you have medical power of attorney, that's control over medical care. Both are legal documents NOT associated with a trust or will. But the person with POA can take money to do things in the best interest of their person. Things can get very shady here if there's money or property that will be left to the heirs and the POA wants to get creative. For example, you could pay yourself for visiting your dad, or paying his bills…something that he didn't specify in POA.

This article from the American Bar Association spells it out nicely.

https://www.americanbar.org/groups/real_property_trust_estate/resources/estate_planning/power_of_attorney/#:~:text=A%20power%20of%20attorney%20allows,creating%20a%20revocable%20living%20trust

Find him a wonderful living environment half way between the two of you and meet with an Elder Law Attorney to help you strategize. Oh, and while you're there, make your own plans for your end-of-life decisions (both sisters).

The POA does not “have to be” the primary caregiver. However, in my case, I would refuse to take on the role of Primary Caregiver without also having full POA. It would be one too many hoops that I would be willing to jump through for all of the moment-to-moment decisions that need to be made for my LO.

I’m curious about comments about levels of trust. Personally, I’d rather not be trusted to take on full caregiving duties if one of my family members deems me not to be trusted with POA. I’m absolutely open to sharing those duties (POA, too). Funny…no one in my family is taking my offer.