My siblings and I do not have the same father, and we've known that since a very early age, yet we do not refer to each other as

I think brothers/sisters usage is quite acceptable. Of course, if I had a brother with no legs....

No, nor is it anyone else's perogative to determine how you think of and refer to yourselves. 

Inaccurate?
With English as it is now, yes, unfortunately.
We need a much larger vocabulary of terms for relatives and relationships.

In the case of half-siblings, there is a physical difference in the genes which has implications for inheritance, health and procreation.
A half-sibling would not be a good organ donor.
If each father has different wealth, each of the mother's children will inherit unequally.
The marriage between half-cousins who do not share a grandparent would be risk-free, being fully heterogenous.

Emotionally the difference between halves and fulls makes no difference in a situation such as yours, Randy.

But in some families, the different fathers can have significant effects.
If the mother has the succession of fathers living with her and each plays a parental role, the dynamics will change hugely. Each man may favour his own child. Each will have a different parenting style, values and rules. Depending on how these changes interact with a child's developmental stages, the effects would likely be huge and unequal.
Even so, many mixed families do work out great solutions, especially among new-agers.

Australian Aborigines have scores of words for relatives, composed with roots, prefixes and suffixes. The meanings define up to four generations in either direction, gender, father's totem, mother's moiety, skin group and clan. With such a small population (approx. 600,000 thinly spread across a vast landscape,) the rules of who can talk to whom and who can marry are designed to prevent the problems of incest. This post was edited by Benedict Arnold at January 3, 2018 6:32 PM MST

Children raised together, whether half-siblings, step-siblings, adopted children or whatever other category there might be, should consider themselves brothers and sisters. That is the relationship that they have. Sometimes, when there is a big age difference or the children are not raised by the same parent (s), they might prefer to use some other term. It's a personal choice and no one's business but theirs.

That's how it is in my family too. My sisters are my father's children and my brothers are my mother's, and none of the children specify how we're related. My family dynamics are all kinds of messed up though. We usually use different names to denote the closeness of the relationship/ lack of closeness with others, particularly between biological parents, biological parents we have contact with, biological parents we have no contact with, and step-parents/ former step-parents. In those cases, distancing language is often used.

It's no one's concern how you refer to your sisters and brothers.  If someone asked for clarification that you were half siblings, you can also simply say we are brothers and leave it at that.  If a doctor is trying to make a diagnosis based on your parents' traits you should make sure she knows if you have a step parent.  Other than that whose business is it?

I grew up with my brother and half brother by another father. We never called each other anything but brother and sister. No halves about it. ;)

When I was a kid I called my half-brother "brother" but I didn't suspect he was really a half-brother until later.  I refer to him as half-brother but I don't see him anymore nor do I ever want to see him.

Call them what you want. It is between you and them.
I have a sister (full), step brother and a step mother.  We refer to him as our brother. I used to refer to her by her first name only when talking about her specifically. Because I felt I would be disrespecting my Mom to call another woman Mom. When speaking about her and my Dad, I would say parents. Once I had children, I began referring to her as Nana, she is their Nana. 
The only time I use the term "step" is if someone needs to completely understand the relationship. My kids were older and shocked when the realized they were not related by blood. 

I think it's perfectly acceptable to call your family anything you want.  Family is family.  

I have a brother that I found out is likely a half.  My mother was never sure and it was a family secret until we were adults.  It didn't change a thing in my mind or his.  

i think its good you all did that