A Trotskyite is simply a follower of Trotsky. It probably started out as a perjorative term because Trotsky and Stalin were rivals, each had their own followers and ideas, and differences. And eventually Stalin had Trotsky assassinated while the latter was in exile in Mexico. (He was staying at Diego Rivera and Frida Kahlo's house, I saw the movie.) That's about the limit of what I know about Trotsky. The word means nothing other than that. I wouldn't be surprised if Trotsky himself never even heard the term.

Trotsky wrote a lot, you might be able to come up with a definition by reading his writings. But another scholar might not exactly agree with your interpretation. I distrust definitions, other than generalized descriptions based on obvious differences. Strict definitions lead to dogma, and dogma stifles thought.

These are just useful definitions, they are not from some dictionary where they are defined in exquisite detail. "Socialism" is like "Democracy" or "Fascism" or even "Capitalism", no two historical examples are identical, and they are by no means mutually self-exclusive. They all cover a lot of ground, and there's a lot of overlap.

If you want a capsule description of my politics, I'll give it to you.
I believe in the private ownership of property, regulated by democratically elected government to prevent abuse of the worker and consumer by those of wealth and economic power. The power of the State must be kept to a minimum as much as possible, but my guiding principle is "In the absence of proper government, the powerful will simply take from the weak." A free market can help, but it is not enough.

In my opinion, phe primary purpose of government is to protect us from domestic criminals, foreign enemies, and the greed and avarice of those who control wealth and property.

My most useful guide to my ideal government is the Declaration of the Rights of Man (1789). Although the American Bill of Rights (1791) ain't a bad start either. I suspect the two documents influenced each other greatly.

In addition to these primary political rights, I believe the State has an obligation to provide every citizen with certain secondary benefits if possible. Among these are education to the limit of his ability to absorb it, medical care, employment, a dignified retirement, and assistance in the event of economic distress. It goes without saying that practical considerations of finance will limit these benefits, it is obvious that we can't all have everything for nothing, and there are practical limits to such aid as the State can provide. But that obligation nonetheless exists, and it must be met as much as it practicably can and the financial resources of the state allow. The amount and limitations of this aid, and the taxation necessary to provide it, will be determined legislatively and democratically.

This is particularly the case of education. Under no circumstances must education be rationed only to those who can afford it. This deprives the ability of every citizen to better himself and distributes it solely to those who are wealthy. That is intolerable.