The nighttime and naptime behavior does sound like pretty classic night terrors (screaming/incoherent, shaking/shivering, happens upon waking up from sleep or between sleep cycles), but having "night terrors" while entirely awake is not usual/normal.
Sudden tantrums with no obvious cause, or sudden severe mood swings in toddlers and pre-schoolers are not entirely uncommon, although not all kids are subject to them to the same degree. I have found that usually there is a 'reason' for the sudden tantrum, although it can be something as small as "I just remembered that I have a toy dinosaur and I want to play with it but it's not in my hand right now" or "I am REALLY hungry all of a sudden!" Sometimes there's no obvious external cause, but the increasingly intense inner thought life of a developing child overwhelms them when they think of something upsetting. So I wouldn't say that the diagnosis of tantrums is impossible.
On the other hand, something feels a bit 'off' to me with the diagnosis of temper tantrums in this case. This is not a medical opinion, but a parenting one: When your child is obviously distressed and uses words like "I hurt", believe them. It seems to me that there is a physical cause of some kind. It may be something as simple as gas pains or teething, but I personally would work from the assumption that -something does hurt and she needs your help/comfort- and not that she's being willful or misbehaving.
If you feel that there is something medical going on, it may help to film the behavior on your phone so that you can show the doctors exactly what it looks like.
Here are some articles about sudden unexplained crying in toddlers: