Accent and dialect are distinct. Maybe it's because I was born in Scotland, spent my teenage years in the English midlands and north, now live in London and watch TV from the states, but I've never met an English speaker whose accent I had difficulty understanding. And that includes all the heavily accented (although generally second-gen) immigrants I've met.
Dialect on the other hand means words I don't know the definition of, and sometimes there isn't enough context to intuit it.
Exactly. But it's obviously not just the accent thats at issue here. If you think about it, and try to do a good impersonation of someone talking in a different accent it has to include a bit of dialect and local grammatical idioms to make it convincing.
I assumed that the original comment about 'foreign accents' also encompassed the grammatical tics that foreign speakers often have. Someone speaking grammatically correct english in a strong accent is easily understood, but it only takes a very little grammatical mistake to change to meaning of a sentence. For example, an Italian colleague who missed out a 'to' and told me 'just a minute, I'll come over your desk' caused some admittedly slightly childish giggling in the office...
Dialect on the other hand means words I don't know the definition of, and sometimes there isn't enough context to intuit it.