I would lean towards no, it does not. The properties (and very name of the feat) would indicate that the additional effect gained at level 9 is in extension to the rules established at level 2 when Unarmored Movement is first gained. A DM might rule differently, but I would lean towards the requirement from level 2 still apply for the level 9 additional property.
No since it's an improvement over the original Improved Movement feature, which work when you are not wearing armor and doesn't specifically remove the need to
Improved Movement: 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.
To reinforce this if you look at the PHB (I only have it here so under sources>PHB) there is no separate entry for the 9th level feature. It is only under the 2nd level feature that Plaguescarred quoted. Unlike the DDB entries which lists it as a separate feature.
Unarmored Movement Improvement
At 9th level, you gain the ability to move along vertical surfaces and across liquids on your turn without falling during your move.
the text doesn’t actually mention an armor restriction. Does this mean a heavy armor wearing monk can still run up walls?
I would lean towards no, it does not. The properties (and very name of the feat) would indicate that the additional effect gained at level 9 is in extension to the rules established at level 2 when Unarmored Movement is first gained. A DM might rule differently, but I would lean towards the requirement from level 2 still apply for the level 9 additional property.
No since it's an improvement over the original Improved Movement feature, which work when you are not wearing armor and doesn't specifically remove the need to
To reinforce this if you look at the PHB (I only have it here so under sources>PHB) there is no separate entry for the 9th level feature. It is only under the 2nd level feature that Plaguescarred quoted. Unlike the DDB entries which lists it as a separate feature.
EZD6 by DM Scotty
https://www.drivethrurpg.com/en/product/397599/EZD6-Core-Rulebook?
DDB does this for coding purposes i assume, but they are one and the same unique feature Improved Movement that gets additional benefits at level 9.