A phobia is exaggerated and irrational by definition.
Exaggeration can vary by personal opinion.
And most people haven't said how they react to a spider when they've called themselves arachnophobic.
Jumping out of a seat, or running away is exaggerated. Other members of my family who aren't scared of spiders would just slowly get up and find something to pick it up with. I can't go near them at all and freak out if they come near me. That's certainly an exaggerated response to them.
And a fear of spiders is irrational. In the UK atleast, there are very very few species of spider that have any form of venom at all, let alone lethal varieties (unless you just happen to be alergic to them). There's a small number that can even bite.
To flee in fear is irrational.
You could say that it's a more rational response, since spiders can be dangerous (so who wouldn't move away from them), but the very same can be said of dogs. Dogs are much more likely to kill a person than any spider is to even bite someone, and a dog bite would hurt more and likely do more damage. But you'd probably consider someone with a complete phobia of dogs to be irrational.
Adding to mine, I also have a phobia of Moths.
Butterflies I'm fine with. I wouldn't necessarily want them landing on me, but Moths I can't stand. The look, the sound, the feel. Their presence scares me, and in the autumn when they all head indoors to try and survive, I feel constant paranoia at them being near me (since you often can't see them hiding in the dark or on fabrics).
I also have a slight fear/discomfort towards caterpillars and woodlice. Caterpillars look nice (or some do) but the feel of them puts me off, and some just creep me out being near them. Woodlice I'm not afraid to see, but touching them sends a shiver down my spine. I can touch and hold them, which I've done in school as part of experiments, but it does weird me out.