etymology - Difference between "help + [infinitive]" with and without "to"

Englishgrammar.org has an article discussing cases where an infinitive does not use the word "to." One case is with many causative verbs like "make" and "let."

I want the water to run

*I want the water run

I make the water run.

*I make the water to run.

It is also the case with "perception" verbs.

I see the water run.

I hear the water run.

The site describes "help" as a special case that can be followed by an infinitive either with or without "to."

She helps me stand.

She helps me to stand.

My question, essentially, is why "help" is a special case that can either ellipse the "to" or not, and what the semantic implications of that choice are?

My hunch is that since one category of other verbs that do not require a "to" infinitive are causative, the choice of whether or not to include "to" has a semantic bearing on whether the subject in the sentence is perceived as a causal agent.

So in the example, "she helps me [to] stand," we would infer from the inclusion of "to" that the subject (she) is less of a causal agent than when "to" is excluded.

It seems possible that there are no broadly applicable implications from the inclusion or exclusion of "to" in an infinitive following "help," that perhaps it is an aberration by way of etymology or coincidence, but I've been unable to find sources that analyze the case beyond identifying it as an outlier.