I couldn't find a command that runs a script on the player.

Any suggestions?

What kind of script is this exactly?

Not plain sure here since I never modded Morrowind, but I think you must run a quest. There is little sense attaching scripts to the player.

Quests, objects, NPCs will start the scripts, and sometimes they will run as 'from' the player.

You are very unclear. I guess you could try to attach it to the NPC "player", but I have no idea whether it will work as you intend.

A Non Player Player Character?
NPCs are all 'others' actors with which the Player can interact and talk, that aren't controlled by humans.
A human controlled character drove by script sounds strange to me.

Yet, the PC can interact with some NPC that have a script that will run having the player as the reference. An object wielded by the player can have some script affecting the player character, an activator can run a script.... A quest is something that can have one or several scripts that the player will walk thru....

I think you mean placing some special ability at the player... something like pushing away anything that get near him enough, for an example...well, this would be an ability ... thing that is like spells, and in some way is almost the same. So you can create such ability, give this ability a script and so give this ability to the player.

hope this helped.

Edit: I think I can clarify a bit more the concept of an NPC 'starting' an script running with the player as reference. You can imagine, for example, approaching said NPC will paralyze the player, or force him to move toward the NPC, or force him attacking the NPC... the imagination is the limit, but such script is not a script of the player... it is a script of that NPC. Or even a quest script.

Morrowind uses a different concept I saw now. It have Global scripts that acts someway as quests in oblivion.
I remember now something about automatic running of scripts being possible only with Tribunal+ expansions. Maybe is about this you want to search.
at CS the command is StartScript name

Anyway I don't know about the particularities of modding morrowind enough to be helping. This thread appeared as I was searching for recent posted ones, at oblivion forums and I got here by mistake. Same like the other. I'll be more cautious from now on.

I know "NPC player" sounded weird, but in Morrowind it's actually correct in some way. In the list of NPC's in the Construction Set there is one NPC that's called "player". This is the player! When you start the game you start as this NPC. Of course it is no longer a NPC then, but you see what I mean, you can make changes to the player character by making changes to the NPC called "player".