I'm sorry you are struggling so badly right now. I can definitely relate.
I'm not advocating the si, but I want to answer your questions to the best of my ability.
I have found that the response to si differs by person and by setting. In my experience, ER staff have seen it more than urgent care centers, and can be a bit more understanding of it being a coping mechanism rather than a suicide attempt. The ER can be scarier, but also a bit gentler depending on the particular staff treating you. I have found that most times, even for only si, they will have you talk to a crisis counselor or have a psychiatric eval. Again, depending on the severity of the cut, the placement, and the staff working, this can go quickly and they discharge you with instructions to follow-up with your treatment providers, or you could get stuck there.
In my experience, urgent care staff don't see as much of it, and will either mistaken it for a botched attempt, or still insist on you getting an eval. The few times I had gone to my doctor's office instead of the ER, she always sent me to the ER for an eval (this had to do with the placement, and my frequently "going a bit too far" by accident because I had been getting less and less relief). A few times, the ER staff sent me home with a crisis plan and a follow-up with my therapist. There were also some times that I was admitted for a few days. It totally depended on the staff and the hospital I went to/was sent to.
There was a time I was seeing an nurse practitioner that knew my therapist and worked closely with her. I was able to go to her for "a quick check" on an emergency basis and she would fix me up, then call my therapist while I was with her. She only ever sent me to the ER once out of about 2 dozen times, and that was only because she misunderstood what I had said.

all that being said, is there any way you can ensure minimizing damage? I know sometimes I have deliberately used a very dull blade. It gave the over-all feeling of cutting (the pressure, the scrape, very little blood), and I was able to refrain from doing too much damage. It also allowed me to be more "violent" in my si (especially if I was angry at myself, or feeling really out of control) without doing much damage. I would only do this at times when I felt I would do too much damage either from having held off for too long and the urges built up too much, or from being too emotionally out of control.

I have also been known to place a phone book or large stack of magazines in front of me and cut them instead. It was also helpful when I felt more out of control. If I got the emotional energy out on the unassuming paper, I would either be ok not cutting, or I would do it much less on myself.

Lastly (or should this be first actually?), can you call you T or a crisis line to talk about what's going on? I know I tend to act first, then talk later, but figured I would throw that out there also. It can be really tough sometimes, and any help you can get re-directing your energies or resisting the urges can come as a blessing. I would hate to see you have to get sutures for a slip up. I would much rather read that you were able to reach out and use alternative coping skills. I know it's difficult though...

Good luck, hope you can find relief without self injury.