A proper ground strap connection should show 1 to 10 megohms between the part that contacts your skin and the ground terminal of the power cord. This resistance is there to prevent electrocution through the strap.
The point of a ground strap is to dissipate static charge. A large resistor does this slowly enough to not "zap" you, but also doesn't turn you into a large ground rail yourself.
There are several considerations for your wrist strap.
A) Safety for you
B) Safety for your fragile components and kit
C) Process, use them correctly
D) Grounded?
A) Your safety
Your wrist strap should include a large amount of current limiting before connecting to a real earth. In practice, this is a 1meg resistor in the strap, and/or where you clip it to, and/or a connector that you plug into an earthed outlet to contact the ground pin.
B) Your kit
There is no point grounding yourself, if the stuff you're working on can float to any potential. Use a conductive sheet, and place all your tools, components, work in progress, on it, and ground the sheet (via a safety resistor) as well. Metal foil, sheet or a tray will do. In industry they tend to use conductive plastic, which is nicer to work on.
C) Process
Having the right equipment doesn't help if you don't use it correctly. When your PI arrived through the post, it was (obviously) not connected to your grounded sheet. At some point, you have to connect them, and it's at that point a damaging charge transfer could occur.
When you connect them, make sure the point that connects first is a grounded point of the PI, a connector shell for instance. Before you unwrap a component from its conductive bag, or pull it from the conductive foam, touch the bag or foam to bring it to the same potential as you (ground).
D) Grounding?
Once you, you tools, your components, your work in progress are all at the same potential, it doesn't matter whether the whole equipotential group is actually connected to ground or not. In practice, it's a whole lot easier to keep track if it is, and as soon as you use a grounded soldering iron, 'scope or power supply you have that earth connection, so you may as well start off with it.
Best Answer
The point about static electricity is that there's usually not that much energy, i.e. charge, building up over a short time.
Therefore, a couple kOhms resistance don't make that much difference - your body voltage would break down over that.
I'd argue that the fabric ones are better for most usage scenarios: they are simply more comfortable to wear, and that makes a difference in usage, and also, humans become sloppy over time – and that'll lead to you being more likely to take off the wristband "earlier".
I'd say: no. Not at all. Ok, it depends on the conductivity and thickness of that lacquer, but assume the following. I'll try to draw a worst case scenario.
You take your angora bunny to your workplace.
For some reason, that bunny gets aggressive/bitey when you don't pet with your wrist, so you pet it with your wristband, and by doing so, you're constantly stroking with the outside of your wristband.
Charge builds up on the wristband's outside. However, that charge is either very small, or directly discharges into the metal part.
So you're fine.
Someone notices you and your pet rabbit at work and fires you, but not without offering the position of chief rabbit safety overthinker in his night-time job, organizing rabbit fairs. You become famous and rich.
Not that bad for a worst-case scenario.