"Formal" Japanese and "honorific" Japanese, are completely different, right?

Question

My understanding is that, strictly speaking, "honorific" Japanese 敬語{けいご} is completely different from "formal" Japanese 丁寧語{ていねいご}. Yet, I sometimes hear even native speakers conflate them:

ex:
「する」の尊敬語{そんけいご} --->「なさる」
「する」の丁寧語 --->「します」
「する」の謙譲語{けんじょうご} ---> 「いたす」

「書く{かく}」の尊敬語 ---> 「お書きになる」
「書く」の丁寧語 --->「書きます」
「書く」の謙譲語 --->「お書きいたす」

formal (not honorific) escalation:「だ」-->「です」-->「である」-->「であります」
honorific (not formal) escalation:「です」 -->「でございまさう」

Strictly speaking, am I right about this? But, is the reality that the usage of honorifics necessarily means you are speaking formally? For years now, I've heard the phrases "honorific Japanese", and "formal Japanese", often used interchangeably. Isn't there a distinct difference?

Indeed, I've read a little Mishima, and he writes formally, but not honorifically. So, there must be a clean difference, and it's just that some people carelessly interchange those two phrases, right?

Answer 1

尊敬語 is when the subject of the sentence is shown respect.

謙譲語 is when the subject of the sentence is being humbled.

丁寧語 is when the addressee is being shown respect.

(Note that the subject is often not explicitly in the sentence.)

From the definitions, it should be clear that it is possible to combine 尊敬語 and 丁寧語, or 謙譲語 and 丁寧語, but not 尊敬語 and 謙譲語 (in a single clause). So, to answer one of your questions regarding なさる & なさいます, the former is 尊敬語, while the latter is both 尊敬語 and 丁寧語.

Regarding how to classify 丁寧語, most people consider it to be a kind of 敬語 (for example, 大辞林), but there are some linguists who do not consider it to be 敬語. In normal discussion I think that it is safe to assume that 敬語 contains 丁寧語.

Formality (e.g., である) is yet another dimension unrelated to showing respect.

Answer 2

I go by the definitions & examples below which I based on the explanations in 日本語能力試験　完全マスター　N3文法.

As Darius says, formality is another neutral form which would used to write academic papers for example which focus on being both concise and precise.

Definitions

尊敬語 : Respectful language: refers to the actions of superiors

謙譲語１: Humble language1: Verb forms to describe one's own action when it affects a superior

謙譲語２: Humble language2: Verb forms to politely/modestly describes one's (group's) own actions (not affecting a superior)

丁寧語 : Polite language: Verbs forms to be polite w/o reference to hierarchy

Examples

尊敬語：

どうぞご自由にパンフレットをお取りください。

では、次に、こちらのグラフをご覧ください。

謙譲語１：

荷物をお持ちしましょうか。

謙譲語２

来月国から両親が参ります。

丁寧語：

客：「すみません。牛乳はどこですか」

店員：「あ、牛乳ですか。牛乳はあそこのパン売り場の隣にございます。」