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I'm interested in the precise definition of the term "radical". The
dictionaries I'm referencing do not always list all components of the
character that could be a radical.

For example, 訓 contains 言 and 川, and I expected that both of those components
would be considered radicals, but the dictionary listings below only consider
the 言 a radical.

<https://jisho.org/search/%E8%A8%93%20%23kanji>

<https://jitenon.com/kanji/%E8%A8%93>

Here's another kanji, 万. This time, 一 is the radical, and the part below is
not consider a radical.

<https://jisho.org/search/%E4%B8%87%20%23kanji>

<https://jitenon.com/kanji/%E4%B8%87>

What makes a component of a character a radical, and what distinguishes that
from a "part"?

In my first example, 訓, both child components are used as radicals generally,
but in the case of this specific character, it seems only the first component
is considered a definitive radical for this character.

In my second example, 万, the first child component makes sense, but the second
part does not correspond to any radical I know of that is used generally, so I
suppose this time it makes sense that only the first component of this kanji
is considered a radical -- I guess?
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kanji radicals
What is the difference between a radical and a part?
In my understanding of the [Wikipedia
article](https://ja.wikipedia.org/wiki/%E9%83%A8%E9%A6%96), it is ultimately
arbitrary what is a radical of a character (as long as it is a part of the
character).

The purpose of radicals is to provide indexes to Chinese characters in a
dictionary. What is defined to be the index for a particular character is its
radical. As such, a radical of a character can be different from dictionary to
dictionary.

For example, a Chinese dictionary 康煕字典 and a Japanese Kanji dictionary 新字源
differ in what is the radical of 飲. The former lists 飲 as 食部4画 and the latter
as 欠部8画 (the Wikipedia article lists some more).

So for your question,

> What makes a component of a character a radical, and what distinguishes that
> from a "part"?

It is dictionary editors' decision that makes a component a radical. The
difference is only that it is defined to be a radical.
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The difference lies in the original Japanese and Chinese terminology in Kanji
studies. Radical corresponds to 部首{ぶしゅ}, whereas "part" roughly means
偏旁{へんぼう}. Your confusion apparently stems from the difference between 偏旁 and
部首.

The first thing you need to know is what 偏旁 is. There is no exactly English
equivalent, at least within the realm of layman terminology. It is usually
translated as "part" or "component" in English. A part or component, as the
words themselves suggest, is what you get when you try and take apart a kanji.
As per [Wikipedia](https://ja.wikipedia.org/wiki/%E5%81%8F%E6%97%81):

>
> 偏旁{へんぼう}とは、漢字の字体を構成する要素の一つで、左右上下内外の部分に分解できる要素をいう。偏や旁を総合して言う。冠と脚をあわせて偏旁冠脚{へんぼうかんきゃく}ともいう。

All the 偏旁 are well studied and usually well defined—although there have been
historical changes, for example 肉->月, and cross-linguistic and regional
variations among the kanji used in Japan, mainland China, Hong Kong, and
Taiwan, but that's outside the scope of this question. So the "child
components" you mentioned are all 偏旁. The study of 偏旁 is historically rooted
and goes back centuries. As Wikipedia suggests, a comprehensive study of 偏旁
was done as early as 1716:

> 現在の偏旁冠脚は、基本的に『康煕字典』（1716年完成）を使っている。

部首 is coded so as to help dictionaries index kanji. Chinese dictionaries and
Japanese kanji dictionaries list and index kanji by what's called 部首. If you
are confused about radicals, 部首 is your guy.
[Wikipedia](https://ja.wikipedia.org/wiki/%E9%83%A8%E9%A6%96):

> 部首（ぶしゅ、英語:
> Radical）とは、漢字を分類する際に用いられる漢字の一部分である。また、それによる分類の、各グループである。部首による分類では、全ての漢字に一つの部首が割振られる。

部首, as used by dictionaries, is a modern thing and it is so designed that
every kanji at least belongs to and falls under the category of one 部首, and
because of that dictionaries can use 部首 to index all the kanji. 部首 as
@sundowner explains is variously defined and coded in different dictionaries
and different countries.

The reason that in a word you only see one radical is that's how it's
designed, in order to avoid confusion and to follow the general rule that one
kanji can only have one set radical 部首. A component may be a 部首 in one kanji,
but a non-部首 component in another. Admittedly this categorization is not
without randomness, arbitrariness, and contradiction. That's why you may see
differences among different countries and different dictionaries.
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