The actual protocol used doesn't matter—many smart devices connect to the Internet of Things using ZigBee, Z-Wave, Bluetooth or another protocol via a gateway which then connects the devices to the Internet.
Many would say that a device which isn't controllable or accessible via the Internet is not an Internet of Things device, however; it's not the means of connection that matters, it's whether it can be controlled remotely through the wide area network of the Internet.
It's difficult to give you a solid yes or no in this case, because various authors have different interpretations on whether the Internet of Things actually requires the Internet to be involved.
Here are just a few examples of answers to whether a device needs to be using the Internet, or whether a local network is part of the Internet of Things:
From the name IoT, we can also learn that the "Things" are connected to the Internet. Accordingly, from the name we can also deduce that the system is not an Intranet or Extranet of Things.
IEEE Internet of Things (2015): Towards a definition of the Internet of Things
In the broadest sense, the term IoT encompasses everything connected to the internet, but it is increasingly being used to define objects that "talk" to each other. "Simply, the Internet of Things is made up of devices – from simple sensors to smartphones and wearables – connected together,"
WIRED: What is the Internet of Things?
It therefore appears logical that the Internet of Things must also run the IP [...] and that all the new clients to this extended Internet, the “things”, must connect to the same network and therefore run the Internet Protocol as well, right? Wrong. Of course, in a perfect world of limitless power on effortlessly miniaturized wireless devices integrated in everyday things, this would be true. But the reality is that the technologies that have the greatest potential, in terms of size and cost, to empower most of the IoT in the short term, can not run the Internet Protocol because they just don't have enough juice to do it.
What the Internet of Things is NOT
I don't think it's surprising to be confused; I certainly can't give you a definition that won't be contradicted by others. Your suggestion of saying a "smart device" definitely seems correct, as the devices are embedded with electronics to be able to interact with others.
Generally, I'd say the following are more common principles:
If the device uses a protocol like Bluetooth and ZigBee to connect to a gateway and then to the Internet, it is definitely connected to the Internet of Things. A lot of smart homes use devices like this, such as ZigBee or Z-Wave light switches connected to a smart hub (like Samsung SmartThings).
If the device is only accessible with a local connection (i.e. on the same Wi-Fi network, or using Bluetooth only, with no control from the Internet), it is less clear whether it is considered to be connected to the Internet of Things.
Often trying to decide whether something fits into the Internet of Things is a bit of a waste of time; there is a lot of buzz and hype surrounding this space and there simply isn't much standardisation yet. You could definitely argue that your "smart thermometer" uses the same principles as any other IoT device, but the sticking point is if it doesn't actually connect to the Internet to be controlled.