Also I was slim for 16 years as a vegetarian, and vegetarian diets used to be considered healthy. Gained weight in the last three years, though - but I eat lots of sweets...
Vegetarians don't have it easy when it comes to losing weight, since some of the most common vegetarian foods can be high in natural sugars and carbs and low in fats and protein. But you certainly can lose weight this way. I usually have a mix of fruits, vegetables, starches, fats and proteins in the meals I cook for myself, or if i'm lazy go for a reduced-fat peanut butter & jelly sandwich on wheat (substitute banana for jelly for a healthier alternative).
It's getting trite at this point but you really should include a steadily-increasing weight lifting regimen to shed fat and gain lean muscle, in combination with the healthy eating. It doesn't even have to be as expensive or hard as the OP describes. My gym has a set of machines that target specific muscle groups, and a numbering system that lets you go down the line using each one in sequence. They also have small instructions for the number of reps and how to change up each rep. This is not the same thing as "lifting" weights, but combined with some actual lifts like the deadlift and squats, can be very effective at growing specific muscle. Which will help you lose fat quicker :)
Basically pick anything that reduces your caloric intake. Yes, some diets might get you more rapid results, but in the end most people fare better with something they can actually live with, especially if it's not just sprinting towards a goal, but keeping that dietary change for a long time (permanently, even). I'd be wary of picking any extreme as your first stop, in any endeavor.
For me, just training yourself of snacks and sweets worked out quite well (I've got one hell of a sweet tooth). I did this mainly by picking 24 consecutive hours in a week where I didn't eat anything (and neither made up for it before or after that). That alone saves you some calories, but for me the main benefit wasn't this fasting, but teaching yourself that you don't have to eat all the time.
I'm sure some see 'paleo' as pseudoscience, but there are definitely people in the community trying to pin down the science.
Although I do guess that the "sole true human diet" factor is contributing to the effectiveness, as people who accept that are more likely to stick with it.
If you're vegetarian for health reasons it might be time to reevaluate. If it's for ethical reasons it just means you really need to sit down and look at the macronutrient composition of the foods you eat. Not eating meat isn't a license to eat starch and sugar.
You absolutely don't need to do the paleo/keto nonsense at all. I eat meat twice a week and barely eat any dairy (just a little cheese once in a while). I eat pasta, white rice and potatoes all the time. I've lost 68 pounds so far as of this morning. No matter how much people want to pretend there is magic involved, or point to hilariously bad books like Taubes' junk, it really does come down to "burn more calories than you consume".
I can't tolerate low carb diets, I feel hungry and tired all the time and end up going nuts after a couple weeks and breaking down and eating a ton of pasta. So I didn't go low-fat or low-carb, I just eat reasonable portions, make sure the food I eat is always satisfying, and exercise. I find it much easier for me to increase the calories I am burning than to decrease the calories I am consuming too far.
And contrary to what spking says, it is not hard at all to get enough protein as a vegetarian. You only need ~55g a day, which I get easily on days where I don't eat meat. On meat days I end up around 90g. If you are doing heavy lifting you'll need maybe 20g a day more than average, and if you do endurance sports you could need up to double the average. But if you aren't doing either of those things, you should easily get enough protein without making any special effort or using supplements.