In this context, “Tivoized” has a fairly specific meaning (conveniently summarized by @codinghorror).
It has nothing to do with how restrictive, or not, a device is in terms of being a dick to the user in the UI, refusing to cooperate if asked for files, etc. On those matters, I have no opinion of Tivo.
It specifically refers to the (admittedly darkly elegant) strategy of deploying cryptography early in the boot process; and sometimes at other points as needed, to enjoy the benefits of GPLed software for your device while simultaneously nullifying the ability of anyone else to take advantage of the freedom you were afforded:
Tivo became the name behind the concept because they were one of the early, linux based, and widely distributed devices to employ the strategy; it’s a lot more common now(locked bootloaders in cellphones, any ‘secure boot’ implementation that won’t let you re-key it; basically all remotely contemporary consoles); the principle remains the same.
Even if you fully comply with any GPL or similar requirements; or use well understood and supported hardware, or both; if your bootloader checks for your signature on the binaries; only software you have blessed can be run, period(barring a specific flaw in your implementation; not uncommon; but not good to rely on). I don’t know what Tivo’s original logic was; whether it was something about copyright; a period when they were selling hardware at margins that would be unsustainable if people could modify it to use 3rd party guide services or just use it as a standalone DVR, or something else; but they got in on the game early, so it’s named after them.
A Tivoized device(unless you know you have a vulnerable one that can be kept from ever updating); is the most aggressively useless hardware in existence for anyone who wants to change the device’s behavior, whether in ways large or small. Reverse engineering a complex piece of hardware isn’t easy by any means; and it often never happens; but factoring some large primes to derive a signing key is “not possible within the bounds of our universe” level difficult; and signature and/or hash verification tends to touch both the kernel and all the various ancillary userspace stuff, config files, etc. so that even the partial modifications you might attempt on something too ill-understood for a full firmware replacement are locked out.
A Tivoized device can be varying degrees of hostile or benevolent; as the vendor’s interests dictate; but the important thing is that it is permanently tied to the vendor’s interests, support(if any); and desired behavior.
