Better to have the exact "servicename" name, so would go for the .eu but buy both. That said, surely there are more options than only .eu and .com?
Options other than .eu and .com are not reliable. .org/.net/.biz are kinda niche and have their own associations (non profit/internet provider/business). All the new TLDs are sort of stupid, and in my (very little) experience, are not indexed well by Google. Countries TLDs like .de, .fr, etc - are very country specific and does not target EU as a whole
I trust .de domains too for the same reasons as that's a kind of default pan European domain mainly based on amazon.de and caseking.de.