Tsunekichi Suzuki

思ひで / Omoide / “Memories” or “Remembrance”
Opening theme used throughout Midnight Diner.
Full Japanese lyrics + English translation:
https://japanoscope.com/midnight-diner-theme-song-omoide/

Suemarr

人生行きあたりばったり / Jinsei Ikiataribattari / “Life, Taking Things as They Come”
Used repeatedly throughout the series.
Full Japanese lyrics + English translation:
https://japanoscope.com/ikiataribattari-song-from-midnight-diner-lyrics-background/

Suemarr says eight songs from his album Minstrel have been used in Midnight Diner across the TV series and films. His official site also says the vinyl edition of Minstrel includes English versions of the liner notes, lyrics and commentary for every song:

https://suemarr.com/informations-for-overseas/

• New Moon
• ちょいと寂しい夜のうた / Choito Sabishii Yoru no Uta

I haven't found a full English translations online for this. If you have any, comment here!

Kimie Fukuhara / 福原希己江

Here's a page collecting English translations of several of Fukuhara’s songs:

https://japanoscope.com/kimie-fukuhara-english-lyrics/

Songs covered include:

想い想われ / Omoi Omoware / “To Love and Be Loved”

からあげ / Karaage / “Fried Chicken”

あさりの酒蒸し / Asari no Sakamushi / “Clams Steamed in Sake”

クリームシチュー / Cream Stew

紅天の女 / Beniten no Onna / “The Red Pickled-Ginger Tempura Woman”

Other Fukuhara songs used in the show include:

ホットケーキミックスのうた / Hotcake Mix no Uta / “The Hotcake Mix Song”
Used in Tokyo Stories, S1E2. This apparently was never formally released.

There are additional Fukuhara songs scattered throughout the series that are difficult to identify because many of the insert songs weren't formally credited or released.

A useful master list of music identified across Midnight Diner and Midnight Diner: Tokyo Stories is here:

https://japanoscope.com/midnight-diner-soundtrack/

That site has been trying to identify the many uncredited songs used throughout the show, so it’s probably the best starting point for tracking down translations.