Let the Germans help.

[viscid hinge]

'Ich habe Lust/Bock auf' it's the same, 'Bock' is more likely used in Umgangssprache.
'Ich möchte...' means I'd like to
'Ich hätte gern' means I'd like to have

the last one is normally used when you're at a restaurant and want to order something 'Ich hätte gerne zwei Cappuccino zum Mitnehmen'

[umbral eagle]

Is it possible to say "Ich möchte" when ordering?

[viscid hinge]

yeah, but you have to add a verb at the end 'ich möchte zwei Brötchen bekommen'

[umbral eagle]

So what is "haben Lust/Bock auf" used to ask for?

Eat?

[viscid hinge]

no, those are used to say that you want or don't want to do something

Ich habe keine Lust darauf, raus mit meinen Eltern zu gehen - I don't want to go out with my parents

[umbral eagle]

So how is it different from "möchte"?

Same?

[viscid hinge]

möchte is I would like to/ I wouldn't like to

Lust/Bock auf etwas zu haben is to want or want not to do something

[umbral eagle]

Thank you very much