What is the correct procedure for removing the anode cap from the CRT without tearing?
You should not be messing with the anode cap unless you have experience with this. The CRT acts as a large capacitor that can retain a high voltage for several days unless it is discharged. The anode cap is often difficult to remove after being in place for years, exposed to ozone and other atmospheric contamination while the volatile organic compounds (VOCs) of it original composition slowly evaporate or break down into other chemistry...
The safest way to discharge the anode of a CRT is to use a length of high-voltage, stranded, high-voltage ignition wire to "short out" the CRT anode BEFORE attempting to remove the anode cap. Be sure the "strands" are actually wire and not carbon-coated plastic!
Use a flat plastic insulated stick, such as a disposable "fast food" plastic knife, to carefully lift one side of the rubber flap on the anode cap, thereby exposing one of two metal wire clips that secure the anode cap to the CRT anode electrode. Strip off about two inches (50mm) of insulation from the high-voltage ignition wire and fan out the strands until only two or three are protruding from the end of the wire. You will use these strands to "short out" the anode terminal of the CRT BEFORE removing the anode cap. Fold and wrap the remaining strands around these few strands and then twist these two or three strands tightly together.
Now strip the other end of the ignition wire and connect it to the outside coating of the CRT as shown in the website that
@Harald Kapp linked you to. DO NOT rely on the chassis for this "ground" connection! Finally, slip the plastic stick under the anode cap far enough to reach the two metal clips securing the anode cap to the CRT. Slide the bare end strands of the ignition wire under the plastic stick until the wires contact one of the metal clips. Leave in place a few minutes and then use your fingers to gently squeeze the two metal clips under the rubber cap, allowing the cap to be separated from the anode terminal. If the rubber cap has "welded" itself to the CRT envelope, use the plastic rod to carefully separate the rubber cap from the CRT glass.
If you are going to make a career servicing CRTs in this manner, you can make a more permanent connection between the strands of the ignition wire and the plastic stick by using a thin layer of Super Glue to attach the wires to the stick, folding the ends back around the end of the stick to ensure better contact with the anode cap. Make sure you don't accidentally insulate the wire with the Super Glue. You could even get fancy by adding an alligator clip to the opposite end of the shorting wire. Maybe add a big red tag explaining what this spiffy "tool" is used for.
The cathode ray tube (CRT) is under very high vacuum on the inside, essentially zero pounds per square inch. The outside is at atmospheric pressure, approximately 14.7 pounds per square inch. Any scratches on the outside surface of the CRT present a potential point of future failure if the CRT is even slightly bumped while handling the heavy glass envelope. You do not want to be anywhere near the CRT if it should implode, because sharp glass shards will go flying everywhere.
There is a safe way to de-commission a bad CRT that involves first placing it completely inside a sturdy cardboard or wooden box. A heavy metal stake is then driven into the center of the face of the CRT. Most of the glass will shatter and be safely contained by the box, which can then be re-cycled or otherwise disposed in a landfill. This is often accompanied by a very satisfying whooomp! sound as air rushes in to replace the former vacuum inside the CRT.