This entire is pure propaganda from top to bottom. It's entire purpose is meant to dissuade people from attempting to use and express their constitutional rights and submit to authority — hence why the entire thing is also one giant argument from authority. We are guaranteed certain rights under Title VII of the United States Civil Rights Act of 1964.

You do not need participate in a particular religion to qualify. In the US, we have freedom of religion; it is not "freedom for particular religions." An individual's "sincerely held" religious belief does not have to be part of a particular religion's position on the vaccine to be considered a valid reason for exemption from getting it. This article — as well as those who support similar views — attempt to make the case that because their Church or group doesn't hold particular views, then they themselves must not hold particular views. The government themselves can make a very strong case in this instance specifically because they openly expressed that they were following their church's views. This is a very important thing to note. This is not a case of a single person expressing their "sincerely held belief" but is instead a case of a group attempting to do so — and it's much easier to disprove something when that perspective.
I hope this is making sense? Essentially, they're not arguing from the perspective of an individual but are instead arguing from the perspective of a group following their religions views.
How can anyone claim they have a "sincere belief" when it's more a belief of the religious group they participate in, which they simply follow? That can easily be discredited or raise reasonable doubt by bad actors.

The best advice I can give is that unless absolutely necessary, you do not disclose absolutely any information on what your "sincerely held belief* would be. It's not their business and they have no need to know, unless they're attempting to coerce or manipulate you. No one else besides you gets to determine what your core beliefs are, nor how deeply you believe in them.
I'm aware it's redundant but I'll repeat it again: You're not required to attend a church or any other such thing for this stipulation to apply to you. In the US we have freedom of religion. This means a religion can essentially be anything that you can adequately explain that you practice or believe (if absolutely necessary to explain) and in which you hold your core beliefs. It does not state we have "freedom to specific religions approved by authoritative sources." Even if you're a Christian, your belief does not need to conform to the authoritative "representatives" of Christianity. They do not determine what your "sincerely held" beliefs are simply by virtue of the positions of religious leaders. Your relationship with God, as well as your interpretation, spirituality, and faith, derived from that relationship, are strictly between you and God — no one else.

I can't stress how important these things are to know if you're adamant in your position on the vaccine. Many companies are being scumbag assholes and trying to screw their employees when they attempt to express their right to refuse. When handing in a letter they'll say that they won't accept it and will instead print up a lawyer created form that has numerous questions that was prepared by their HR departments and legal teams. This is a kafka trap to screw you. Do not fall for this. The reason they're doing it is to get you to answer certain questions so they can later use it against you. It's most certainly a type of legalese to trap you so they can fire you without worry, or, at the very least, have a strong case in court against you should you choose to pursue it in court.
If necessary, you can even film their refusal to accept your exemption letter or attempt to coerce you. If they choose to fire you for not filling out their questionnaire — after simultaneously refusing to accept your exemption letter — you will have a strong case in court for discrimination and retaliation. If worse comes to worse, you can also simply fill out only the relevant parts that leave zero risk of them potentially scrutinizing and using your answers against you. You can then write next to it "I do not feel comfortable disclosing that information."
 
In the end, don't simply take my word for things. If it's important to you then don't be lazy. Do more research on the topic and get a better understanding of your rights before it becomes too late and you're put into a situation where you suddenly have to crash course learn important information. Consult with a lawyer if necessary. There are plenty out there who would likely do it for a low cost or even for free if it's simply a few questions. Send an e-mail to a Civil Rights attorney to get a better grasp on your situation in your area.
Just don't be complacent. Make sure you're well informed and have a full understanding of why this is a sincerely held belief for you. Bookmark studies and data; save authoritative articles; accumulate things that you feel help form the beliefs that you have and then correlate them to yourself spiritually.
 
To end this, I'm going to post an example of a simple exemption letter that can hopefully help some people. There are many others out there as well. Some people purposely try to go into great detail on their letters, some prefer not potentially providing any information that may be used against them. Personally, I would go with the latter and only do the former if I deem it necessary — but that's just me.
 
I hope this helps at least one person:

I, ______________________, declare religious exemption from the COVID-19 vaccination mandate. Under Title VII of the Civil Rights Act of 1964, I attest to having sincerely held personal religious beliefs which will not permit me to abide by this mandate. Additionally, I am not willing to describe these personal religious beliefs for the reasonable fear that if others have knowledge of them it may engender discrimination against me and/or my family.

I am aware that reasonable accommodations may have to be made regarding this declaration of religious exemption, and am willing to discuss accommodations that would be necessary and reasonable in order to come to an agreement. I will finally also expect this letter to remain confidential, and its contents to only be known to those for whom there is an express need.