You can be a vegetarian in many degrees.
Some don't eat meat
Some don't eat meat, fish, or shell fish
Some don't eat meat, fish, shell fish, or any dairy.
Do what you feel comfortable with and what works best for you.
And no one is perfect, you may have times that you slip from a craving, doesn't mean you are a horrible person, just try and get right back into your preferred diet.
I'm right there with you. I don't eat meat at all, or shell fish, but I tell you what...I love fish and chips and I'm not going to say I haven't slipped on that a few times a year. Especially summer time, I just crave halibut and chips. Yes, I feel like a huge toad afterward, but I just try harder the next time to say, no.