In the first part of this answer, I’ll try to show you how (and: how differently) behaglich and gemütlich were used in the past, and how they developed into what they are now.
Hopefully, this will help you develop a feeling about when to use which word: this is difficult if all of your examples come from modern dictionaries.
In the second part, I’ll comment on the theory in your question.
Part 1
Behaglich and gemütlich have different origins, and they retain different meanings until today:
gemütlich is related to the gemüt, one’s disposition or nature.
The etymology of behaglich is not really apparent (it could be related to the old Germanic word hagan: protect, foster, nourish).
A bench can be gemütlich or ungemütlich to sit on, but that is not an absolute quality as it depends on your situation.
Whether it is comfortable is not an inherent quality.
However, the context of reaching that spot, sitting down, taking a break, having a picnic, enjoying the scenery etc. can make the whole situation gemütlich or behaglich, and this is where the two words overlap.
The »Deutsches Wörterbuch« by the Grimm brothers (DWB) offers numerous literary examples (many more than quoted here) for behaglich, but none of them supports using behaglich to describe a piece of furniture or other specific object:
Logau:
wer redlich ist im herzen und mit dem munde frei,
der wisse, dasz bei hofe behäglich [comfortable] er nicht sei.
Goethe:
sag nur wie trägst du so behäglich [even-tempered]
der tollen jugend anmaszliches wesen? (here and below, toll means crazy)
fürwahr sie wären unerträglich,
wär ich nicht auch unerträglich gewesen.
Rückert:
baue nach lust dein feld,
nach deinem bedarf dein haus,
und sieh auf die tolle welt
behaglich [untroubled] zum fenster hinaus.
For example, a room with a fireplace could be described as gemütlich or as behaglich, and here, both words point in a similar direction:
behaglich doesn’t necessarily mean it’s snowing outside, and gemütlich doesn’t necessarily mean there’s a comfortable armchair next to the fire, although both could be true.
Instead, both words here could mean that the whole setting is calm and inviting.
The DWB defines gemütlich (this is the relevant definition) as:
3) genehm, lieb, willkommen, dem wunsch und sinn entsprechend u. ähnl.
What does this mean?
It means that the word is used mainly to describe how one feels about something:
Charlotte v. Stein:
wenn es ihnen gemüthlich ist [if it is convenient], so grüszen sie auch Göthe von mir.
Goethe:
es war mir gar nicht gemüthlich, dich heute zu verlieren, und so hab ich mich deines Fritzes bemächtigt [I was displeased about not being able to see you today, so I met your Fritz instead]
In particular, it mentions the attributive use:
b) auch mit attributiver, also flectierter verwendung des adj., obschon selten
as with behaglich, the examples are manifold, but the only example referring to a specific spot is by Goethe:
traten wir in den garten um das grab des edlen greises (Gleim), dem nach vieljährigem leiden und schmerzen, thätigkeit und erdulden, umgeben von denkmalen vergangener freunde, an der ihm gemüthlichen stelle [the spot that the deceased liked] gegönnt war auszuruhen.
From following the shift in the last example, we can deduce the contemporary usage.
Note how the subject (emphasized) changes:
antiquated:
Es war ihm gemüthlich, an diesem Orte zu verweilen.
still antiquated:
Die Bank ist mir gemütlich.
modern:
Die Bank ist gemütlich.
The meaning of the word gemütlich has changed from describing whether something accords to one’s gemüt to being a positive adjective to describe objects.
Its old meaning remains in the phrase »Mach’s dir gemütlich«, and we know how that famous sentence from »Life of Brian« was translated:
Sit down. Have a scone, Make yourself at home. You klutz!
Setz dich. Nimm dir 'n Keks, machs dir schön bequem. Du Arsch!
Therefore it shouldn’t surprise us that the »Digitales Wörterbuch der deutschen Sprache« (DWDS) lists »bequem and behaglich« as the first meaning of gemütlich.
Part 2
You wrote:
gemütlich describes the comfort of a situation / a mood / atmosphere / something external.
Yes.
The old usage of gemütlich relates to one’s gemüt, and that is internal, but it expresses the gemüt in regard to something external.
Nowadays, it is used to describe objects, but the origin of the word still shines through, since whether or not something is gemütlich is not absolute but depends on your gemüt.
Now, it seems like behaglich has some overlap here, but more refers to the FEELING that somebody has in this situation.
Yes, there is an overlap.
But as you may see in the examples for behaglich (first part of the answer), substituting gemütlich for behaglich wouldn’t work in these cases.
You further asked if the following was true:
- gemütlich describes objects, behaglich does not. Ein gemütlicher Stuhl, not a behaglicher Stuhl.
- Both can describe atmospheres. Eine gemütliche Wohnung, eine behagliche Wohnung.
- Something is "behaglich" if it leads "gemütlich".
Strictly speaking, I believe you are correct about point 1, but nowadays there is an overlap in meaning, particularly in the spoken language.
I agree with point 2, but again, this is umgangssprachlich.
Imagine you are invited to a friend’s for the first time.
You see his room and say one of the two sentences below, and, colloquially, both should work:
Du hast es hier aber behaglich.
Du hast es hier aber gemütlich.
You will hear gemütlich more often than behaglich, and the connotations of the two may be a bit different, but it will also depend on the person you talk to, on local speaking habits etc., so I wouldn’t say there’s a hard rule.
Finally, you asked:
As I attempted above, maybe behaglich is comfortING, and gemütlich comfortABLE?
As to gemütlich and comfortable, you are correct.
However, as far as I understand comforting, it means to cheer up or to console: both imply that you’re not well to begin with.
This is not the case with behaglich.
You can feel perfectly at ease and still think a room or situation is behaglich or pleasant.
If I had to name a German word with a meaning close to behaglich, I’d suggest heimelig, which the Leo dictionary translates as homey, homely, haimish, if that helps.