I cut ZoE's nails with people toe nail clippers. They're cheap and I throw them about when they start to dull, so I always have a sharp set. I've handled her paws since she was 6 weeks old and she has no issues with me cutting her nails at all.
A friend gave me a nail trimmer/filer similar to the one you pictured, for Christmas. ZoE doesn't like it at all!!!
Just checked...I have the "pedipaws" kind. Horrible. Just the sound of it freaks her out, but then as soon as it goes against her nail...pure craziness (imagine holding your nail against a belt sander to file them....
).