Do I have arachnophobia?

All my life I've been scared of spiders, since my mother made up the "biting spider" that lives in the basement and dies on Christmas day to keep me from seeing the gifts in the basement. But I'm not sure if I qualify for having the phobia. I heard that just seeing a picture of a spider would make an arachnophobe have like a seizure. This morning I saw a spider in the basement with leg span of about a 50 cent coin and screamed at the top of my lungs, ran upstairs, and called my mom to hurry home to kill the tarantula in the basement. I know that is quite irrational, but I did not have a seizure or anything. And I actually study spiders all night some days on the computer, interested in the different kinds and the most venomous. But whenever I see one, I freak out and if it's at night when I'm alone downstairs, there are issues. The most recent time I saw one scurry across the floor, I sprinted upstairs to my bedroom and shut the door and went to sleep in fear, without even leaving my room to brush my teeth. Before that, I spent an hour trying to kill a tiny spider, which I eventually did, using 2 five pound weights, a guitar hero controller, and a flip flop. Although this happens to me with most bugs, spiders are the only ones that make me scream (with the exception of large silverfish and house centipedes). Do I actually have arachnophobia? Or is it just a small fear that generates huge screams? Just note that when I am fleeing a room, I don't grab anything but myself, not even my phone.