What's a positive phrase to say that I quoted something not word by word

Paraphrase is the perfect word.

To paraphrase Churchill, we will fight them everywhere.

Or

Paraphrasing Churchill, we will fight them everywhere.

There are plenty of examples at Lexico.

Or, less formally, you could use gist:

He said, "We will fight them everywhere." That, at least, was his gist.

Or

He said, "We'll fight them all over the place," or words to that effect.

Paraphrase is perfectly good here, and probably the most natural choice in most written contexts e.g.

Paraphrasing Oscar Wilde, "truth isn't pure or simple"

You could also caveat the word "quote" with suitable hedges. The suggestions in the question ("keeping with the sense", "in spirit", and "conceptually") don't feel right to me though, and seem somewhat of an oxymoron together with the word "quote". I would expect something like "loosely" or "roughly" e.g.

Loosely quoting Oscar Wilde, "truth isn't pure or simple"

Indirect speech is also generally assumed to be paraphrased, and often summarised e.g.

Oscar Wilde said that truth wasn't pure or simple

In especially informal contexts, quotative "like" indicates that speech is paraphrased, often with subtext drawn to the surface, and often for dramatic or humorous effect especially with a slightly exaggerated tone e.g.

Oscar Wilde was like, "that's rubbish, truth isn't pure or simple!"

There are also lots of phrases that can be used with direct speech to show that it's been paraphrased, but these generally only work with the verb "to say", and not "to quote" (which has the strong implication that the words that follow are verbatim). They can also often occur both immediately before or after the quote often with slight changes in the wording of the hedge e.g.

Oscar Wilde said, "truth isn't pure or simple", or words to that effect

Oscar Wilde said words to the effect of, "truth isn't pure or simple"

As others have said, "to paraphrase" is perfectly apt. An alternative with a slightly broader meaning is "to borrow", as in:

Borrowing from Seneca, luck is a function of preparation and opportunity.

As spotted in Tristan's answer, "summarise" could also be used in some contexts.

Indirect speech is ... often summarised e.g. ...

I've'd [sic] to face this in a book I am writing, chronicling our community (Uganda Asians/Indians)'s history. I have to bring in materials from written texts. If it's a quote it's a quote and quote marks have to be used. But I also have to summarize/paraphrase often in which case I say at the start of an extraction: >>In their own words but often extrapolated, except when an exact quotation is used in which case I use quote marks.<< "Exact quote" itself is a difficult thing to adhere to. Often you add something in which case square brackets are used and omit something in which case ellipses are used.

You can always use "If I recall correctly" which is frequently abbreviated to IIRC so examples might be:

• If I recall correctly Fred said "Yabba Dabba Who"
• Fred said "Yabba Dabba Who" (IIRC)

The other common phrase is "Words to the effect of..." e.g.: Maggie said words to the effect of "they can make an effort to find work".