While I agree with V2Blast that the purpose of the item seems to be not letting people just Misty Step away, the RAW is very confusing about defining what exactly extradimensional movement should mean. Personally, I wouldn't call teleporting from one place in a dimension to another in the same dimension extradimensional.
Some Spell entries in the PHB seem to indicate that teleportation is not an extradimensional mean (to move), thus not an extradimensional movement. For example, the Spell Hallow:
Extradimensional Interference. Affected creatures can’t move or travel using teleportation or by extradimensional or interplanar means.
The awkward thing is that Teleport (the spell) can't be used to other plane anyway, so it wouldn't make much sense even mentioning it if the purpose was to only restrain people from teleporting to another dimension, unless effect of spells like Plane Shift are indeed considered teleportations, which I consider possible as the wording is very similar:
Plane Shift: You and up to eight willing creatures who link hands in
a circle are transported to a different plane of existence.
Teleport: This spell instantly transports you and up to eight
willing creatures of your choice that you can see within
range, or a single object that you can see within range,
to a destination you select.
Also, Plane Shift can be used with Teleportation Circles, so it might be considered a teleportation method.
Anyway, the RAW is unclear on defining these terms, there is nothing on Sage Advice to make it clearer, so...
Up to the DM.
But, unlike Spells, Magic Items usually serve a specific purpose in the game. You, as DM, are usually using them for a purpose or giving them to the players so they can do something. Just make sure the item serves that purpose. I would think that magic items are one of the least important things to follow by rule. Unless you are rolling the treasures and end up giving them this item. Anyway, I don't think allowing or restricting teleportation with the shackles is a matter of balancing, so you can choose what is better for the plot and fun.