What is the difference between food allergy and food sensitivity?

Help me understand whether this could be a food sensitivity or a food allergy:

tl;dr: I get joint pain in my wrists, a rash on my hands, and a red, itchy neck when I eat gluten. I tested negative for celiac disease, but it and thyroid disease run in my family. I’ve been eating gluten-free for almost two years. I recently developed the same neck and wrist symptoms when I eat dairy, so I’ve given that up as well. Is this a food allergy, or a food sensitivity? And I should probably be worried about leaky gut, right? What are the chances I'll continue to lose more and more food groups?

Background:

My mom was diagnosed with celiac in 2021 or 2022 in her late 50s or early 60s. At least one cousin and aunt also have celiac and thyroid diagnoses.

I thought I had dodged all the problems until late 2022. I started getting a red, itchy neck. I also had some issues with runny stools and urgent bowel movements. I read that many first degree relatives can have asymptomatic celiac disease, so I decided to get tested eventually. However, I didn’t have a primary care doctor at the time and was too busy.

February-May of 2023 was probably the most stressful period of my life, as my workplace was undergoing a hostile takeover, I was one of the last members of leadership left, and I was job searching and moving halfway across the country.

The itching was getting worse, but I knew if I stopped eating gluten before getting tested, the test would not be accurate. Then, during the worst of the stress, my wrists started hurting. For a week it got worse and worse, then the pain started traveling up my arms. At that point, I went to Lab Corp and got the celiac test. It came back negative, I cut out gluten, the wrist pain was gone within days, and my mild digestive issues went away. When I ate gluten for a day, and the symptoms immediately returned.

Since then I figured I have Non-Celiac Gluten Sensitivity, but I haven’t pursued a diagnosis, and sometimes I wonder if the test was accurate, since I wasn’t eating that much gluten at the time.

When I have had gluten by accident, I get small red dots all over the backs of my hands, my wrists hurt, and/or my neck itches and turns red. Once I thought, “maybe I’m not sensitive anymore” and ate a breadstick, and my hands had the red rash for a week.

Recently I started to notice my neck itching again. This was the first symptom I noticed with gluten. I didn’t think I was getting glutened by mistake.

I cut out dairy as an experiment, and the itching went away within a few days. When I had some dairy, the itching came back and my wrists started hurting. This happened even with gluten-free cornbread containing a little buttermilk.

I’ve now cut out dairy, but I’m concerned that I might have ongoing gut issues or “leaky gut,” and that other food groups might be next. I’m also concerned about eventually developing celiac disease—if I don’t already have it—or other autoimmune diseases.

I’m also confused about what to call this, and I haven’t found much online. Some reputable sources say that a food allergy is very dangerous, involves the immune system, and can cause anaphylaxis. I’ve never come close to having breathing problems from food that doesn’t seem to fit. But the websites said a food sensitivity or intolerance mainly involves the digestive system, like lactose intolerance, which doesn’t seem to fit either since I have symptoms in my skin and joint pain.

What do you think is happening, and would it be considered a food allergy? I don’t think my primary care physician is very knowledgeable about this, since she told me I could be tested for celiac without eating gluten.

Appreciate any input!