Is there a commonly used name for a "local ground" or "local common"?
There is no accepted terminology, but most people use similar terms.
I've never heard the term local ground, usually a more descriptive term is used, like what the ground is attached to. The thing I usually do is differentiate grounds if they need to be physically separated, if they don't then just label them ground. Usually chassis ground or mains ground and analog and digital grounds need to be separated. (although many designs do not need separate analog and digital grounds when designers incorrectly think they do)

Grounds fall into two categories as follows: (1) safety grounds and
(2) signal grounds. The second category probably should not be called
grounds at all, but rather returns, and it could be further subdivided
into either signal or power returns. If they are called grounds, they
should be referred to as ‘‘signal grounds’’ or ‘‘power grounds’’ to
define the type of current they carry and to distinguish them from
‘‘safety grounds.’’ Common usage, however, often just refers to them
all as grounds.

Source: Chapter 3: Grounds Electromagnetic Compatibility Engineering
In isolated designs, I have referred to grounds as GNDA GNDB GNDC (to keep them separated in the design)
Another common one is GND and GNDE to differentiate the PCB ground and earth ground or mains/chassis ground. Mains and chassis grounds typically use different symbols.



simulate this circuit &ndash; Schematic created using CircuitLab