The vignette states that:

Instead, one need only run nbinomWaldTest to re-estimate MLE coefficients – these are necessary for apeglm – and then run lfcShrink specifying the coefficient of interest in resultsNames(dds).

We give some examples below of producing equivalent designs for use with coef. We show how the coefficients change with model.matrix, but the user would, for example, either change the levels of dds$condition or replace the design using design(dds)<-, then run nbinomWaldTest followed by lfcShrink.

Which suggests that it is sufficient to replace the design or overwrite the condition factors.

However, it also states:

You should only change the factor levels of variables in the design before running the DESeq2 analysis, not during or afterward.

Which seems contradictory to the above:

cnts <- matrix(rnbinom(n=1000, mu=100, size=1/0.5), ncol=15)
cond <- factor(rep(1:3, each=5))

dds <- DESeq(DESeqDataSetFromMatrix(cnts, DataFrame(cond), ~ cond))
resultsNames(dds)

[1] "Intercept" "cond2vs1" "cond3vs1"

# does not work
dds$cond = relevel(cond, 2)
resultsNames(dds)

[1] "Intercept" "cond2vs1" "cond3vs1"

# works
dds$cond = relevel(cond, 2)
resultsNames(DESeq(dds))

[1] "Intercept" "cond1vs2" "cond3vs2"

And what a user (like me) may incorrectly read from the first paragraph as the "suggested" way:

# did not work when I tried:
dds$cond = relevel(cond, 2)
# I forgot to save the result here:
nbinomWaldTest(dds)
# thus this line was raising an error
lfcShrink(dds, coef='cond_3_vs_2')

Error in lfcShrink(dds, coef = "cond3vs_2") : coef %in% resultsNames(dds) is not TRUE

Is there really a problem? When you relevel, try using characters to specify the reference instead of a number.

E.g. if you want the reference level to be "2", you should use

dds$condition <- relevel(dds$condition, "2")

I'll edit the second quote you have above to say this is one exception to the design-change rule (just for re-arranging coefficients).