The red cover protects the positive terminal; if you look in good light, you’ll see it has a ⨁ embossed in the top. Unless you’re replacing the battery with a new one, you shouldn’t need to open or remove the cover or disconnect the positive terminal. As long as at least one terminal is disconnected, no current flows through the battery. The cover is there to keep the negative and positive terminals from being shorted together, as might happen if a metal tool were inadvertently dropped across them. The part highlighted in red is the nut that tightens the battery state sensor assembly around the negative terminal, not the terminal itself. Perhaps a photo (revised from one I posted earlier) would help: To disconnect the negative (–) terminal, use a 10 mm wrench or nut driver to loosen the nut marked with the red arrow, and then gently lift the battery state sensor assembly with battery current sensor holder, highlighted in blue, up and off of the negative terminal post and move it to one side, so no part of it touches the post. Reconnecting is the reverse; the Repair Manual (more info) says you’re supposed to tighten the nut to a torque of 5.4 N·m (55 kgf·cm, 48 in·lbf).
I wonder, if you loosen that one nut (flagged with red notation), is that whole assembly then free to lift off, be tucked away, clear of the negative post? Also, would some of the components, say the "battery current sensor" device, reduce the need for disconnection during long term storage?? Some cautions, not sure how critical they are:
The horn sounding is probably the alarm system being upset. Use your remote to "unlock" the car if the horn starts sounding. You may have to press "unlock" twice to reset the alarm. If the battery disconnect device fits (doubtful with all the new negative post sensing devices) it would also trip the alarm. It's getting funnier and funnier. Now you can't even disconnect your battery without a Toyota mechanic present with his programming tool?
Some of those initializing items might be overkill, who knows. But yeah: I changed my battery, now the car is broken... If they're not kidding, a jump pack with a tpms cable memory-saver is a good ploy. I could search the manual for those FIVE items, the damn links (of course) are broken in my copy. See if there's some non-Techstream workarounds. Not exactly Toyota's finest hour if there aren't workarounds. I suspect Toyota's blowing smoke on those items, 99 times out of a hundred the car continues to work just fine after a battery disconnect. But who knows...
There are several systems listed, but really only two initialization steps, explained previously. No special equipment is needed.