Some can be identified by cover details. There are may be notes only belonging to the first edition. Clear example is Magical Mystery Tour where later editions had many faces blanked out because public figures did not want to be displayed on the record cover. Same story with Rolling Stones "Some Girls". Another clear example is Led Zeppelin "In Through The Out Door" with bunch of different cover variations.
Many shaded dog releases would mention on the cover "previously released as (on different serial number)".
There are also labels that only did re-issues such as Arkiv re-issued Deutsche Gramophone
Apparently there are and there aren't simple ways to ID 1st pressings.