There is no "open vs concealed" carry in Texas with respect to knives.
There is only "legal vs illegal". The terms refer to whether the knife may be legally carried in public, i.e., off one's own property or vehicle, or on private property owned by other for which you have permission to be, i.e., you are not trespassing.
To be a legal knife, the criteria is simple - the blade from tip to guard/bolster (or where the guard or bolster would be if the knife in question had a guard or bolster) must measure less than 5.5 inches and be single edged.
So as long as the knife BLADE is less than 5.5 inches and has a single sharpen edge, it is legal to carry it in public anywhere in Texas, except where prohibited, e.g., court houses, schools, private businesses with their "employee policies".
The preemption law which went into effect last September 01 nullified all city ordinances regarding the public carry of knives, so various city ordinances were invalidated. There are some ordinances still in effect, such as municiple courtrooms. I am not sure if the preemption bill had any effect on Tyler's prohibition of knives on public transit or not.
With respect to machetes, they fall into the "illegal" knife category as they obviously have blades longer than 5.5 inches. However, you can carry them in your vehicle. I have one in the truck behind the seat and one in the trunk of all cars because if you need one NOW, you can't be having to drive back home to get it. If you need it for work, it is legal to have your work tools in your vehicle. When you arrive at your work site, you have permission to be there to do work, so you may legally possess and use your illegal knife to legally perform the work.
You just can't carry your machete down the street, waving it like an idiot. Or any knife for that matter, because that will fall into the category of "brandishing", which is a different law all together.
To the OP, unless the church in question has a "no knives" policy properly passed by its "church council" (or whatever the denomination calls their governing organization) and publicly posted in a prominent place, there is no prohibition on carrying a legal knife of any kind.
I EDC, literally, paired fixed blades. Even to church. MY church fixed blades have 3 to 4 blades and are of the slim "bird & trout" variety. Outside church, 5.25 inch Kabar shorties. No one says a thing at church to me, but then I was baptized in that church 61 years ago and serve as one of the pastor's assistants during Communion. On a given Sunday, out of 100 adult worshipers, there are 15 open carry and probably triple that in pocket/purse carry knives. Even my 80 y.o. mother has a pocket knife in her purse. A Buck 284 that I bought her because she wanted a knife to carry when she saw how often I used one AT CHURCH to cut things.
