The pre-Civil War Black Codes did indeed restrict the rights of African Americans in some Northern states. They were not always strictly enforced. Most of those laws were repealed in the mid-19th century, before or after the Civil War. The fifteenth amendment ended voting restrictions in the North and what remained was laws against interracial marriage. That's not to say that discrimination ended, just that the situation you described didn't exactly apply after the Civil War.
In the South after the Civil War restrictive Black Codes were being introduced to replace the old Slave Codes that had controlled African-Americans and restricted their rights. It's hard to see what Lincoln may have had to do with Southern states imposing segregation and Black Codes. I'd say that the very fact that Black leaders were free to come to the White House and discuss the plan with Lincoln suggests that conditions in the North weren't as oppressive as you claimed and that Lincoln wasn't committed to forcible relocation either.
Yes, I’m well aware the Black Codes were something else. What I was getting at was “voluntary” might be what somebody says. It might be what is in the legal code. That does not mean that was the reality. As the Black Codes in the North demonstrate or as official laws on the books saying Blacks could vote in Pennsylvania for example demonstrate is, that was not the reality. Make it impossible for Blacks to earn a living and they will leave “voluntarily”. Have any Black person aware that terrible things will happen to them if they do try to vote and they won’t try it as was the case in Pennsylvania. You can’t come along 150+ years later without knowing the reality on the ground, read the word “voluntary” in somebody’s statement and grasp what the reality was without knowing the context.
Lincoln wasn’t committed to forcible relocation. Of course if local laws and conditions made things difficult if not impossible for Blacks and this spurred many to “voluntarily” leave, he’d have been just fine with that - and yes Martha, the Black Codes were very strictly enforced in the Northern states which wanted to keep Blacks out.