Yes, eating fish but no other meat is considered pescatarian. BTW congratulations on your transition! You are doing much to help your health, the environment, and animals. There is nothing bad about that.

That said, you would not be vegetarian if you consume fish. But think of it this way. Are you trying to live up to a label, or does this label convey a greater message to you? I consider myself "vegan", but it is not an end in itself. The reason for being vegan for me is my concern for animals, the environment, and the health of our planet, but strongest is the stance against animal exploitation and cruelty, and living a vegan lifestyle is a step towards the goal of reducing and eliminating these things and a stand I make against the need to exploit animals to thrive. It is something I can do on a personal level to help reduce suffering. And I am also an animal rights activist who leaflets and spreads the word as much as I can. For me, the label of vegan or vegetarian is not so much about the rules or sacrifices or limits, but living a life of compassion, health, and symbolic gesture of refusing to be a part of a system of factory farming, animal exploitation, and environmental destruction. Veganism in itself does not even encompass all that one can do to avoid these problems. You can be vegan but still use way too much plastic, litter, or consume products like coffee that involve the exploitation of child labor. However, being vegan in the sense of being more aware of nonviolence and harmony has made me so much more aware of how I live my life on every level and in every detail, and how this in turn affects others. In other words, I try to focus on the greater purpose and not get bogged down by the rules. Veganism is a distinct definition, for sure, as is vegetarianism, and does carry rules to define it, but it is about so much more.

I wouldn't condemn or look down on you for eating fish. You are doing more than most people as far as your efforts to be healthier and cause less harm. Perhaps this is a stepping stone for you. Some people take years to transition to fully vegetarian or vegan. Some never fully transition but continue to eat and live in a way that works for them long term and sustainably while doing the least harm possible to others. It is a balance. So if this is what you can handle right now, then do not feel bad about your choice.

There are many issues with fishing and the harm and cruelty towards these sentient creatures. Farmed fish are especially suffering overcrowding in tanks and being fed a diet that is unnatural to them (soy and grain because it is cheap vs. seaweeds and smaller fish for those that eat other fish...which also translates to lower omega 3s for the human or animal that eats these fish). If you eat fish, I would at least go with wild caught, and try to avoid fish that are known to have a higher mercury exposure in them. Try to choose fish that are not exploited on the highest level (cod is a fish that is being terribly exploited and their numbers are declining rapidly, and tuna is terribly exploited and overfished, not to mention shrimp). Purchasing from private local fisherman (or catching your own fish) might be a little better than from large commercial fishing companies that use huge nets that drag along the ocean floors and kill many other sea creatures along with fish.

I am not an advocate of eating fish and don't want others to think I am encouraging it, but I do understand that some people are concerned with how to get omega 3s in the diet such as DHA or EPA that can be difficult for vegans and vegetarians. Vegans can get ALA by eating a variety of nuts and seeds and also foods such as avocado. There is controversy about the ability of ALA to convert to DHA in vegans and vegetarians. But there are many long term vegans who have lived successfully without direct sources of DHA in the diet. Google "long term vegans" and you will find a link where hundreds of long term vegans share their experience. I do supplement with vegan DHA that is derived from algae, the source that fish get their DHA from in the first place. I don't necessarily feel I need to, but I do it as an extra precaution. I have been vegan for 2 years and 10 months now and am still quite healthy. My Mom on the other hand attempted to go vegan but can not eat whole nuts/seeds due to diverticulitis, and also has an intolerance to gluten so her diet was very limited as a vegan. She lost a lot of hair due to lack of omega 3s in her diet. It stopped once she started supplementing with DHA but she still found it hard to stick to being vegan and also includes fish in her diet. I can't judge her because I have not been in her shoes, but I do know that she does not have the healthiest diet to begin with and relies on eating out a lot, and she is not as health conscious or passionate about animal causes as I am. But if this is what works for her, it is a far cry than the way she ate before, consuming copious amounts of dairy and chicken and red meat. She is still causing far less harm to others and is eating in a way that is helping her.

As far as low carb, I am not a fan of low carb diets. Carbs are not evil, but it is how you choose to include them that counts. Refined flours and sugars and pasta are nutrient deficient, but whole grains, fruits, and vegetables are nutrient dense, filling, low calorie, and satisfying. I would rather consume natural fresh fruit than bland food that requires a ton of artificial or processed sugar to make palatable. I eat oatmeal with just berries added for example (frozen or fresh) and it is much more satisfying to me than adding tons of sugar. Protein is also not hard to get on a vegan diet and the amount of protein we need has been exaggerated in the western diet. I am not a nutrition expert armed with facts off the top of my head, but I am well aware that there are many vegan athletes (for example Scott Jurek who is an ultra marathon champion and Matt Danzig who is a UFC fighter) and healthy people who get enough protein without resorting to animal foods. My sources of protein include a lot of beans, tempeh, whole grains (millet, wild rice, bulgur wheat, oats, quinoa to name a few), vegetables such as broccoli and other leafy greens, nuts and seeds in small quantities, even nutritional yeast provides a fair amount of protein per tablespoon with relatively few calories. If you look at all the other minerals and nutrients that are in plant based foods along with their protein content, see the whole picture instead of isolating any one macronutrient, you get a better idea of why plant based foods on average are healthier and less harmful to health than animal based products.

I hope this helps. And good for you for doing your research to understand the whole picture and to guide you in making the best choices!