We were in just the same position as you, I had posts pages long on here, people trying to figure out if he was OT or UT or what was going on. Has your little one always been low sleep needs? He seems to be getting so little sleep. Listen, once my little boy started to go by the Groclock (it took about two weeks to get him to a 'normal' wake up time), he was getting more sleep at 24 months old than he was at 18 months. He would do 12 hours reliably every night, and a decent nap.
But yes for sure there is crying, just like there always in for sleep training.
And it all relies on how you respond when he wakes up at 5am.
We didn't use the storybook either. Here's what we did. We explained to Stan that if he looks at the clock and sees the moon, then it is night time and mummy and daddy are sleeping. If he sees the moon, his job is to stretch, grab his lovey, pull up his blanket and go back to sleep (we role-played this). If he sees the sun, that means it is morning and we can go downstairs for breakfast. Simple instructions, no room for confusion.
So he was regularly waking at 5am (had been since he was 10 months old, I was very tired). He was in a BBB and we had already put a gate up at the door because he would compulsively get out of bed at bedtime. I set the clock for 5.30am. The first morning, he wakes up, comes to the gate, screaming and crying because I'm not coming. That's OK. I had explained to him that mummy is asleep. This isn't like leaving an infant to cry it out, this is a toddler having a tantrum because he's not getting his way. Our bedroom was next door so I would periodically call out to him (calmly), "The moon is on your clock, mummy and daddy are asleep. Go back to bed now." At 5.30 am I went in, pointed out the sunshine on the clock, told him how happy I was to see him now that it's morning, lets go down for breakfast.
The next morning, he screamed for about 20 minutes but did quiet down. At 5.30 I went in and said how he did so well to wait for the sunshine.
The next morning, he cried for about 10 minutes, did go back to bed (but didn't go back to sleep). Awesome!
After that, he didn't cry any more, didn't get out of bed and more. I started to move the clock by 10 mins every couple of nights and pretty soon we were at 7am.
I remember one time I HAD to pee right at 5am, and the bathroom was right next to his bedroom, and I was so sure he would wake up and want to get up... I tiptoed past his room and looked through the 2" crack in the door and saw him wake up, look at the clock (moon), and he said 'Oh dear!' and lay down and went back to sleep. I could hardly believe it! I swear I would have paid £250 for that clock if I had known how well it would work. It's like a clock. Literally. You or I wake up at 4am, we look at the clock to know whether to get up or go back to sleep. This is the same thing for the kids.
But, you know it doesn't work magic by itself. You have to be 100% consistent that the day NEVER starts before the sun shows on the clock. And, it may be that he never goes back to sleep in which case you will need to look at routine etc. But at least YOU will be able to rest and get up at a decent hour.