Let's explain them: the first flag depicts the coat of arms, also symbolized in second flag. It depicts a skeleton (or the Death) on a stone with a castle in left. The real origin of the meaning is unknown, but it's popularly said to represent the castle of the 2nd Earl of Ulster, Richard De Burgh (1259-1326), being the skeleton from one of his knights and cousin, starved to death on castle's dungeons. In the chief (top of shield) there's London coat of arms (sometimes with a harp on center), added when city changed its name from Derry to Londonderry.
The field of both flags is a crimson, representing the crimson flag, firstly flown during the Siege of Derry (1689) as symbol of defiance. Sammy's flag contains badge with a oak leaf and a stronghold, taken from a memorial in Guildhall square. The oak represents city's origins and name (Doire means "oak grove" in Irish). The bastion is a reference to the city's insurmountable walled city (hence city's nickname "maiden city").
Based in this flags, I made my own proposal to Derry's/Londonderry's flag:
In the first flag, the coat of arms. I didn't use the London banner of arms to avoid somebody concluding that city has any formal relationship with the English city. The field is crimson, the same shade of flags above. I use the same oak leaves from Doire team in GAA (Gaelic Athletic Association). I'm not still 100% satisfied with the flag, but still think it's reasonable. I tried a half-half version of flag, instead of a "cantoned" one, but that didn't work, for me.
Thanks for reading! Any kind of comment is welcome.
I recommend the visit to Sammy's post about Derry/Londonderry flag: click here.