RAW, armored monks can move along vertical surfaces and across liquids
Unarmored Movement is unambiguous.
Starting at 2nd level, your speed increases by 10 feet while you are not wearing armor or wielding a shield. This bonus increases when you reach certain monk levels, as shown in the Monk table.
At 9th level, you gain the ability to move along vertical surfaces and across liquids on your turn without falling during the move.
The first benefit only works without armor and shields, while the second benefit only works during your turn.
They are distinct benefits with distinct restrictions stated in distinct paragraphs.
If the second benefit were subject to the armor and shield restrictions I would expect the second paragraph to somehow reference the first. For example:
At 9th level, this bonus grants you the ability to move along vertical surfaces and across liquids on your turn without falling during the move.
Is the RAW reading legitimate?
If the RAW reading does not defy our expectations significantly, then I find no reason to stray from it. For the reasons below, I don't think that Unarmored Movement defies our expectations enough to discredit the RAW reading.
Other similarly-structured features
There are other class features that provide two distinct benefits with a similar two-paragraph structure. For example, the barbarian's Feral Instinct and the ranger's Land's Stride just to name a couple.
Unarmored Movement is unique in that the second distinct benefit comes at a later level, but that does not imply that the second benefit is subject to the first benefit's restrictions. If anything, it implies that the two benefits are even more distinct.
Other monk features
Most of the monk's features work while wearing armor and wielding shields, so it's reasonable for the 9th level benefit to also work in such circumstances.
It's also worth noting that the 9th level benefit is implicitly weaker if you are wearing armor or wielding shields as you won't be able to move as far along vertical surfaces and across liquids.
Other features with misleading names
The fact that the feature is called "Unarmored" Movement is of no importance as there are plenty of features and abilities that defy their own naming. For example, the Pact of the Blade feature benefits all melee weapons, not just blades. The errata also agrees with this point when discussing mage armor:
I find it confusing that the mage armor spell is named that when it doesn’t count as armor. Some spells and class features have figurative, not literal, names. The text of the spell or class feature explains what it does. In this case, mage armor (PH, 256) surrounds the target with “protective magical force”; the spell doesn’t provide armor.