There are three basic social types in dogs. As previously noted they are born (inherited), not made. A given group can be all of one type; there does not need to be an alpha present (nor will some other dog turn into an alpha).
Alpha: they know their place in life. They never feel a need to assert dominance (fight, bully, defy authority), with dogs or humans. They have a lot of initiative and can be "too much dog" for a human who doesn't need a dog that wants to work, but give them the slightest bit of leadership and they will follow you anywhere.
Beta: these are the ones that fight, bully, disobey because they can, and otherwise try to exert dominance. Short-man's complex in a dog. Unfortunately this type is what the average person thinks is an 'alpha' but they are not. Betas always defer to any alphas that are present (including humans, which in my observation also sort into these three basic types). Betas come in a range from top-ranked totally-hotshit-all-the-time to submissive-acting sneak-biting fence-fighters, who only bully from behind the safety of a fence. (In a fight, the lower-ranked beta always loses.) These are typically the dogs that give non-assertive owners trouble, and require a lot more effort over and over because they're all about defying authority when they think they can do so. Train via the old-fashioned methods where you are the boss, end of discussion, and they'll generally be fine (and there are a lot of betas among high-end working dogs); but this modern-day feel-good and treat-based training encourages them to bully their owners (which is why today's pets exhibit so many behavioral issues that were seldom seen in the past). Betas tend to pick on anyone they perceive as an underling... including their owner.
Nobody: these are like an alpha, minus the initiative. They are usually ignored by alphas, and by most betas. (A few betas will attack nobodies.) Their reaction to being jumped by another dog is "Huh??" and they don't even think about 'owning' toys or food. They're the easiest for inexperienced owners to handle, because they don't argue and are happy just to follow along. By the same token they don't make the best working dogs, because they just don't have that extra go-gettum component.
Alphas and betas can be hard to distinguish as puppies, because some of the spectrum doesn't express until puberty. But you never want a puppy that refuses to lick your face (you may have to blow in a nobody's nose to get it started, ie. give it permission). Face-licking is strongly tied to desire to interact with humans. But you want moderate face-licking, not one that tries to scrub your face off. And never one that turns its head away and refuses to lick.
As a rule the alpha puppy on its back in your lap wants to lick your face with some enthusiasm, put its feet in your mouth (this is a distinct mannerism), and is never done interacting with people. But it doesn't force its way when you want it to stop (at least if you're halfway assertive).
Meanwhile the beta may nip your nose or or try to get away and is really more interested in pleasing itself -- would rather go play, not sit here quietly like the boring human wants. A low-end beta may just lie there stiffly and turn its head away from you (refuse to lick), but don't confuse that with a nobody who is waiting for permission to lick and is trying real hard to not show dominance. The nobody won't obviously turn away like it wants to leave.
And the nobody just lays there like "Hi, are you going to eat me? Okay!" and maybe does some tentative face-licking, tho it may be all raring to go when it perceives that you want it to do something (here, that is to lie quietly on its back in your lap). Basically, they naturally wait for permission from the leader, which makes them real easy to get along with.
It sounds like you're getting progress since the older dog is starting to wait for permission. Permission is a big deal here. The leader grants permission; the underling doesn't get to just take it. This applies to everything -- food, toys, who gets to go out the door first and who walks in front on the street (always the leader). YOU are the leader; neither of the dogs is.