All of the words mean approximately the same, with slight variations in nuances. 修正、更正、改正share the character正，which means "right, correct," so these three all mean to make something previously wrong correct. 修改literally means "to edit," since whatever was written before may not have been wrong. You're just changing it. However, in everyday use, it does have the meaning to "correct" Finally, 批改has a very strong sense that a teacher is correcting a student's work. The other verbs do not have this sense.

我也想知道【是否】这六个词都是名词和动词。很遗憾，汉语的特点是单词在词性转变时没有明显标志，尤其是动词转变为名词，基本都是靠经验了。 vreb： 他【修改，修正，更正，改正，纠正】了文章里的错误。 noun 他对文章里的错误做了【修改，修正，更正，改正，纠正】。

修改，及物动词。 如：修改作业。 修正，不及物动词，如：对你的答案做出修正。 更正，和修正几乎一样。 改正，即是及物也是不及物动词，如 改正错误。又如：有则改正，无则加勉。 纠正 ,和 改正一样，多用于书面、政治语。 批改，这个词和前面几个不太相同。前面几个词都有“正”字，意思就是说，往好的方面的、正确的方向去做。而批改没有正确的意思，很可能本来错误的东西，经过批改仍然错误。而且这个词局限性很大，只能跟文章、作业等书面材料这些名词搭配。如：批改论文。

可 none, 亦(也)可 verb. Apart from your name cannot be a verb as a common sense in English as well. to put 的 at the end is called Adjectives Such as, 修改的,修正的，更正的，改正的，纠正的 ,批改的 文章article or stuffs. Ha I am not a Chinese teacher, sorry.

"能给我说明区别，我会很感谢" sounds like an output from the google translate