My really old wonder: "defined by" v.s. "defined as"

Question

I understand the following sentence:

The rule was defined as ~~~~ by John.

It can be shortened by omitting "as ~~~~" or "by John", like

The rule was defined as ~~~~.

The rule was defined by John.

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Now, my question is what is difference between "defined as" and "defined by" in the context of Mathematical papers or textbooks.

I saw many times they are used interchangeably, e.g.,

The penalty function is defined by

~~~~~~~~~~~~~~~ Equation ~~~~~~~~~~~~~~~~~~~, (1)

where ~ is ~~~~~~~~~~~~, ~ is ~~~~~~~~~~~, and ~ is ~~~~~~~~~~~~~~~. (variable description)

and

The penalty function is defined as

~~~~~~~~~~~~~~~ Equation ~~~~~~~~~~~~~~~~~~~, (1)

where ~ is ~~~~~~~~~~~~, ~ is ~~~~~~~~~~~, and ~ is ~~~~~~~~~~~~~~~. (variable description)

I guess that both are correct but the meaning is slightly different. What is exactly different???

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Also, some people write

where ~ is ~~~~~, ~ is ~~~~~, and ~ is ~~~~~~~.

but the others write

where ~ is ~~~~~, ~ ~~~~~~, and ~ ~~~~~~~. (Only write the be-verb in the first one.)

Answer 1

From my experience,

as is used for a definition, whereas
by is used for constraints or characteristics

For example,

The speed of sound is inversely proportional to the altitude of an airplane

In the equation, altitude is defined as the height of an airplane in flight

The possible range or values of altitude are defined by the aerodynamics of the airplane.

In particular

The equation is bounded by ...

talks about the limits of the equation, and is usually not described as

The equation is bounded as ...

In your example

The rule is defined as ... by John.

Seems to be a mixed metaphor, since the by is used to express ownership and not necessarily a property of the rule, unless possibly authorship by John is a determining property.

HTH

Answer 2

In defined as {X}

X is the definition.

In defined by {Y}

Y can be a person or entity issuing the definition, or it can be a circumstance or quality that is of integral importance to the thing, making it what it is:

An athlete is defined as a person who engages in a sport.

An athlete is defined by Webster's as "a person who is trained or skilled in exercises, sports, or games requiring physical strength, agility, or stamina".

A world-class athlete is defined by extraordinary skills, strength, and stamina.

Disaster-status is defined by the inability of the stricken community to emerge from the catastrophe without outside help.