What is electrical insulation tape officially for?

Four ways electricians use black vinyl 'electrical tape':

Wrap around an outlet or switch, to cover the live electrical terminals, before stuffing the outlet or switch into its electrical box. This prevents surprises in later opening the (possibly live) box, and protects against contact with the bare ground wire in that box. Other 'live' metal (like big splices) gets more elaborate treatment, but taped twisted-wire splices are sometimes encountered in elderly wiring.

Repair loose insulation (old cloth especially) when reworking an old device. Sometimes even covering a nicked or scorched bit of damaged insulation, if replacement is not feasible.

Identifying different wiring use: if you run a black/white/green cable to a distant switch, the white wire is not 'neutral', but 'switched hot', so is given a wrap of black tape to mark this variant function. Green (or bare) cannot be given a different use, it can ONLY be ground.

Pulling wires, like in a conduit, with a steel 'fish tape'. You twist the wire into the fish tape's end, then cover the edges of the assembly with vinyl tape so it doesn't catch. At the end of a big job, there's a lot of short twists of wire with black tape, and maybe pulling lube, in the discard bin.

The primary purpose of electrical tape is to insulate wires after they have been spliced in an approved manner. The US National Electrical Code permits splices to be made in various ways:

110.13(B) Splices. Conductors shall be spliced or joined with splicing devices identified for the use or by brazing, welding, or soldering with a fusible metal or alloy. Soldered splices shall first be spliced or joined so as to be mechanically and electrically secure without solder and then be soldered. All splices and joints and the free ends of conductors shall be covered with an insulation equivalent to that of the conductors or with an identified insulating device.

So if you really want to, you can wire your house by twisting the wires together soldering them and taping them with electrical tape. I have actually seen this in old houses. But nowadays you will find it easier, quicker, and cheaper to use readily-available wire nuts. (A wire nut would be a combined "splicing device" and "insulating device".)

Electrical tape is still widely used when splicing larger conductors which are too big for wire nuts. Often splice bolts are used. These are bare metal clamps which bite down on both wires. Once the bolt is installed it is wrapped in multiple layers of tape. You can often see these splices at the service entrance where the wires from the street join the cable leading down to the meter.

The reason electrical tape may not be used to splice a lamp cord is that splicing of such cords is not allowed at all (NEC 400.9).

In addition to the excellent answer from Whit3rd, some other uses for vinyl electrical tape are:

Insulating low-voltage splices and connections.

Securing wire or cable to a stationary or supporting beam or post or cable.

One of my hobbies is doing professional Pyro and Fireworks. We use e-tape by the dozens and dozens of rolls in a single show. In addition to all of the other uses previously mentioned, we'll use tape to secure large Roman candles to their racks, tape bundles of small cakes together so as to make them more stable and not able to tip over, dozens of other similar uses.

For What It's Worth, early electrical tape was made of cloth and often referred to as "Friction Tape". Most houses and many commercial buildings were wired with twisted copper connections which were then insulated with friction tape. Sometimes, those twisted connections were soldered before being taped up - but not always.