Two 1955 D Lincolns

Both look like former Encased Cents.

Yup.
John1

Encased Cents have flattened rims, concavity to the edges and often show buckling of the fields.
Here is what a coin can look like if it has been enclased in a ring that had a design around the border. Submitted by Kelsey Olson at the Michigan State Numismatic Society Convention Spring 2012.



Here is an Encased Cent with no design around the border. The coin if removed would look like yours. Submitted by Vaughn Pierson at the Michigan State Numismatic Society Convention Fall 2011. Coin is also RPM-001 / FS-501.

A telling sign of these once Encased Coins is the "pulley-like" edge the coin assumes after being crimped into the holder.

Also, because this crimping flattens the coin's rim, its diameter increases proportionate to the rim's size.

The above left, 1988 cent has such a "pulley-like" edge and its flattened rim makes it seem larger than the comparison coin; a normal size Lincoln Memorial cent. At first, the 1988 was thought to be a " Texas Cent" and presumed to of had been beaten with a mallet while lying betwixt pieces of leather. This technique allows the beaten coin to be enlarged without leaving signs of the PMD being inflicted on the coin. Nevertheless the 1988 cent, and your coins as well, show the post-mint damage incurred by encasing coins in coin holders like the examples shown above. Cheers.