"onto" versus "on to"

My English teacher at school was adamant that on to was always two words, a position which is acknowledged by ODO:

The preposition onto written as one word (instead of on to) is recorded from the early 18th century and has been widely used ever since, but is still not wholly accepted as part of standard British English (unlike into, for example). Many style guides still advise writing it as two words, and that is the practice followed in this dictionary. However, onto is more or less the standard form in US English and in the specialized mathematics sense. Nevertheless, it is important to maintain a distinction between the preposition onto or on to and the use of the adverb on followed by the preposition to: she climbed on to (or onto) the roof but let’s go on to (not onto) the next point.

In British English, it’s always safe to separate on to; in American English, onto would appear to be acceptable in almost all circumstances.

If onto can be one word in the examples given, one has to decide whether on can be classed as an adverb with to as the preposition, or whether onto is an acceptable preposition in its own right.

In Burchfield’s New Fowler’s Modern English Usage he quotes Fowler from 1926:

Fowler (1926) added ... examples of on used as a full adverb before to and therefore written separately: We must walk on to Keswick; Each passed it on to his neighbour; Struggling on to victory. In He played the ball on to his wicket, he judged that “as He played on could stand by itself, it is hard to deny on its independent status”. It should also be noted that They drove on to the beach would normally mean “They continued their journey until they reached the beach” but could also mean ”They drove their vehicle to a position on the beach”; whereas They drove onto the beach could only mean ”They drove their vehicle to a position on the beach”.

With hold on, the on has Fowler’s “independent status”, and there is no sense of movement as there is with driving “onto a beach”. Indeed, the act of holding on to something is precisely to steady oneself and prevent movement! Hold on to is therefore appropriate.

In the second example [“grabbed onto the cushion”], with the verb grab on the to is needed (so on is not independent), and there may even be a sense of movement with a sudden taking-hold of the cushion. Grab onto might therefore be appropriate.

That said, customary forms in different dialects of English may dictate a different use. Grab onto looks decidedly wrong to my British eyes, conditioned as they are, even though I’ve argued fairly successfully from Fowler [a respected British authority] that it’s reasonable; and hold on to may look decidedly wrong to American eyes.

TLDR: People spell this two different ways because they pronounce it two different ways. And trying to convince people to spell it "hold on to" when they pronounce it "hold onto" is a losing battle, so you might as well not even bother trying.

I think I've figured out what's going on here. Namely, I think I've figured out why writing he held onto the rope is so common, and why people are arguing about it so much.

Consider the examples:

I drove on to the next town.
I drove onto the pier.

In these sentences, I pronounce on to slightly differently from onto: onto is a single word with the first syllable stressed, while in on to, the two words which have roughly equal stress.

However, in the sentence

I held onto the bannisters,

I pronounce onto in the same way as in I drove onto the pier, despite the fact that hold on is a phrasal verb. However, some people pronounce this sentence as if on and to were different words. Listen to Forvo.

I would guess that, in general, people are distinguishing between on to and onto by their pronunciations. You're not going to get people to change their pronunciation. And trying to get people to spell hold on to in a way they don't pronounce it is clearly a losing battle.

Which is correct? Clearly, for pronunciation, both alternatives should be considered correct. But in writing, I would suggest using hold on to. While normally, I'd suggest going by the pronunciation, that doesn't help here because it's pronounced both ways. And there are enough people who get upset at hold onto that you might as well make them happy. However, I also think it's pointless trying to get people who pronounce it hold onto to spell it hold on to. So, to avoid arguments, both spellings should be considered correct.