Levitate on self is up and down movement with 20 speed
The confusion on which type of movement you gain through the levitate spell comes from this line in the spell description (PHB 255):
The target can move only by pushing or pulling against a fixed object or surface within reach (such as a wall or a ceiling), which allows it to move as if it were climbing.
This line suggests that any time a creature is levitated it can only move using a climbing speed. However, if the target is brought down to the ground I don't rule it that way as a GM and here's why:
Firstly, there's this line at the end of the spell description (emphasis mine):
When the spell ends, the target floats gently to the ground if it is still aloft.
By stating here, "if it is still aloft" the description clearly indicates that the spell can end with the target aloft or the target not aloft. If the target is not aloft that means it is on the ground and it can walk, it no longer needs to pull itself and use the climb speed. It also suggests that the target can fluctuate between being aloft and not aloft, which means you can walk, levitate and then walk some more.
Although his rulings are certainly not the same as RAW, I think Mike Mearls provides some access to RAI and he agrees with this point in this tweet about a Genasi casting the Levitate spell:
@mikemearls if an air genasi casts levitate on self. Can they still walk normally on the ground and float at will or can they only float?
they can walk while on the ground #wotcstaff
Additionally the spell allows you to raise or lower yourself up to twenty feet:
You can change the target’s altitude by up to 20 feet in either direction on your turn. If you are the target, you can move up or down as part of your move.
If you go up 20 feet and then you go down 20 feet you are on the ground. If you are on the ground you can walk, and if you continue to concentrate you can continue to levitate yourself up or down as part of your move on each turn.
The definition of movement speed is, "How far you can move on your turn with a certain type of movement." Or as stated on PHB 189:
On your turn, you can move a distance up to your speed.
You can levitate 20 feet as part of your turn, which means your Levitate speed is 20.
Whenever you switch movement types, subtract the distance you've already moved from the new movement's speed
There are two references to multiple movement types in the PHB.
The first is on PHB 190 under the heading Using Different Speeds, which you've quoted in the question. It also provides this explanation:
For example, if you have a speed of 30 and a flying speed of 60 because a wizard cast the fly spell on you, you could fly 20 feet, then walk 10 feet, and then leap into the air to fly 30 feet more.
This rule is applied if you have two types of movement each with their own speed: If you have more than one speed.
In the example you provide you have two different types of movement, each with an indicated speed, 20 Levitating and 30 Walking. Whichever you use first is subtracted from the type of movement you use next, and if you use one of them again the current total is subtracted again from the total available for that movement type.
Assuming you start your turn and you walk 30 feet, you can no longer levitate on that turn because your levitate speed less the distance you've already walked is less than zero: 20-30=-10, and per the rule above leaves you with no more movements. However if you walk 10 feet, you can levitate 10 feet onto a balcony and then walk another 10 feet. The explanation is that you walked 10 feet (20-10=10) so you can levitate 10 more feet. You have now gone a total of 20 feet and can no longer levitate (20-20=0) however you can walk because 30-20=10, so you have 10 feet of walking remaining.
Note that the order in which you use your movements matters and you'll need to think it through to gain the maximum advantage of both speeds.
The second reference is on PHB 182, under the heading Special Types of Movement:
Climbing, Swimming and Crawling
While climbing or swimming, each foot of movement costs 1 extra foot (2 extra feet in difficult terrain), unless a creature has a climbing or swimming speed.
(This section, strangely, says nothing about crawling)
This section provides a specific rule in reference to 3 specific types of movement: Climbing, swimming and crawling, unless you have a specific climbing or swimming speed. This rule is only for PC's who want to climb or swim and do not have these movement types as a trait with their own speeds. This rule is not relevant to the specific situation you describe because in your case you have two movements, each with their own speed.
However this rule would indeed apply if you levitate into the air and then decide to move yourself by pushing. In that case, as the spell description states, your speed would work the same as climbing. In that case whatever distance you "climb" would count as 2x your walking speed and would subtract from either your levitate or walking speed, depending on what you choose to do next.
Is this a clear RAW ruling?
It is clear to me that the text supports this position, yet I can see why someone else would rule differently. If this is something you see coming up in your campaign it would be worth checking in with your GM about how they would rule it.