Have to disagree with the vowel sound you use in BIRD is a long vowel. The long vowel "i" sound is the "i"sound in "find," "mine," "grind," "like," "tribe."
The difference between the ir and er is, indeed, the accent. Probably has roots to British pronunciation (e.g., Director = British "dye-rector" as opposed to American "da-rector".)
An interesting tidbit here, I think, is that the short "i" vowel in "bird" has no phonetic purpose: A "silent" vowel. Take the vowel out of "bird" or any of those "vowel-r" examples and you can still correctly pronounce the word!