There are two different things that might help you understand about chicken feed. First, how it is made.
They grind all the ingredients together and mix them. That is called mash. To make pellets, they wet that mash, push it through a dye, and quickly dry it. To make crumble, they slightly crush the pellets. It’s all the same thing nutritionally since the ingredients are the same, but depending on the machinery you use to automatic feed or the feeders you use, one might work better than the other.
Then they have the different analysis. If you look at the label, you will see different percents of nutrients like protein, fiber, calcium, salt, and other things. That’s the difference in Starter, Grower, Developer, Layer, Flock Raiser, and all the other things they call them, the actual ingredients in them. It doesn’t matter if that if mash, crumble, or pellets.
The big difference in Layer is that it has really high calcium content so hens can use that extra calcium for their egg shells. Young growing chicks should not be fed Layer because that extra calcium can harm them, especially their liver and kidneys. Once they are laying age, it doesn’t matter.
The only significant difference in all the others is just percent protein. The general idea is that Starter has a high protein content to get them off to a good start, then you can switch to a lower protein feed to get their bodies to better match maturity rate with body growth. But it really doesn’t matter that much which of these you feed at which stage. They’ll do OK on any of them. The main thing is they don’t need the higher protein feeds after they get off to a good start and those higher protein feeds are more expensive. Why spend extra money to buy them something better than they need when they do fine on the less expensive stuff? But there are plenty of people on this forum that do that and it does not hurt their chickens.
I occasionally switch between crumble and pellets. Sometimes they don’t especially like the switch but I’ve never had one starve to death. If they get hungry they’ll eat it.
I hope this helps you a bit. That person at the feed store sounds like they don’t quite know what they are talking about.