Getting back on topic as kindly requested
(apologies for the wild digression earlier).

The docstring for `RR` currently reads:

```
sage: RR?
Type:           RealField_class
String form:    Real Field with 53 bits of precision
File:          
 /opt/s/sage92b6/local/lib/python3.8/site-packages/sage/rings/real_mpfr.pyx
Docstring:
   An approximation to the field of real numbers using floating point
   numbers with any specified precision. Answers derived from
   calculations in this approximation may differ from what they would
   be if those calculations were performed in the true field of real
   numbers. This is due to the rounding errors inherent to finite
   precision calculations.

   See the documentation for the module "sage.rings.real_mpfr" for
   more details.
```

which very adequately

- starts with the words "an approximation to the field
  of real numbers using floating point numbers"

- ends with a reference to the sage.rings.real_mpfr documentation,
  which itself, as accessed by `sage: sage.rings.real_mpfr?`,
  clearly formulates what we are dealing with.

So I think we're mostly fine, but I understand the need
to clarify the string representation.

The first sentence in the `RR` docstring suggests we could replace
the current string forms:

```
sage: RealField(8)
Real Field with 8 bits of precision
sage: RealField()
Real Field with 53 bits of precision
sage: RR
Real Field with 53 bits of precision
```

with the following ones:

```
sage: RealField(8)
Real Field approximation with 8 bits of precision
sage: RealField()
Real Field approximation with 53 bits of precision
sage: RR
Real Field approximation with 53 bits of precision
```

or if we need something shorter:

```
sage: RealField(8)
Real Field 8 bit approximation
sage: RealField()
Real Field 53 bit approximation
sage: RR
Real Field 53 bit approximation
```

I'm also fine with other suggested solutions.

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