You do not need a purpose-built E-bike for riding in a rain. A proper E-MTB can take a lot of nasty weather punishment, including weeks of daily rain, snow, parking at freezing temperatures, riding over few cm of wet snow, pulling with the engine against strong wind and the like. You also need front and rear lights, same as in the night, and the mudguards are the must. Rear one is especially important, without it your back gets dirty. And I do not know if the engine would handle the full submersion, maybe better not.
If you need to commute over 20 km one way or over 300 m uphill, the battery life needs attention. You may look for E-bike with the easily removable battery that you could charge separately at work under your desk. In a university (that was high uphill) nobody cared. In a private company now I do not need this (and have no courage to ask), but maybe permission would be given.
I have never tested a tricycle or recumbent trike, leave alone electric one, so cannot say much about them. They looked for me unsuitable for narrow and generally bad paths that under success may be closed for car traffic, but other say they are not.
What you need is to prepare yourself.
To dress for electric bicycle in cold weather, makes sense to visit also a motorbike shop. The clothes (pants, gloves, jackets etc) there are often warmer, do not count on the driver producing lots of heat and are also heavier. But in cold rain on E-bike it is really almost never too hot and with the engine assist you have less requirements that it should be featherweight and not constraining. Of course, do not buy that already feels like medieval armor and the pants look very funny while at work, you need to change.
As correctly said by @Criggie, most ideal would be to have a dry and heated place to store a bicycle while at work or even charge it, but just bringing inside is problematic as it gets dirty on the way. Otherwise would be great if the employer provides at least a place to hang wet clothes where they could dry. In the worst case only bring gloves and helmet cap inside. Waterproof pants usually do not get very wet all way through (by idea and design) and can stay in a hard (plastic) box on the luggage rack of the bicycle (textile bags do not handle a day of rain regardless that they producer says). Shower is optional for E-cyclist, it is always possible to adjust the level of assist so that you do not get hotter than when travelling by bus.
You need rain-proof gloves, pants, jacket and cap under you helmet. You do not need some special shoes but at work you may need another pair of dry shoes. With all this in place, I actually enjoy riding in a rain.