I got confused by learning the wrong Grammar terms. What are the real ones and where to learn them?

When I started to learn English, I did it with half a century old Soviet book called The Practical English Grammar by Izrailevich and Kachalova (Практическая грамматика английского языка. Израилевич и Качалова).

There are sections in the textbook about the Syntax and the Morphology of the language. As far as the Morphology section is concerned everything’s looked spot on so far.
But I started learning the Syntax which explores not the parts of speech as units but a whole sentence. And there I began to see problems.

It’s just that some terms in the book which break sentences into components don’t seem to align with the ones people use around here, on this site. But, oddly enough, those terms try to be analogous to the Russian terms in the Syntax of the Russian language. So my point is that the textbook tries to make me learn the English Syntax through the Russian terms.

I’ve done some googling and found out that all the other Russian English learning resources offer those Russian-like terms too. But the real English speaking world seems to use a different taxonomy, as it were.

On this site I find terms such as Prepositional Phrase, Complement, Adverbial Phrase and many others, but I don’t find them in my textbook. Instead, I find there the following:

I (The subject) gave (the simple predicate) the letter (the direct
object) to them (the prepositional indirect object).

I (the subject) gave (the simple predicate) them (the indirect object)
the letter (the direct object).

A boy (the subject) feeds (the simple predicate) the dog (the direct
object) with pleasure (the adverb of manner) in the yard (the adverb
of place) every day (the adverb of time).

To my surprise, people here get confused when I use the term the indirect prepositional object referring to to them.

I’ve never seen anybody use the term the simple predicate or simply predicate referring to the verb of a sentence here. If someone used it, they meant the whole sentence, I guess, after the subject.

So, I’ve been wondering if anybody knows what’s wrong with the terms which have been taught in Russia so far? Why, at least, some of them differ from the ones used by native speakers of English? And by which resources I can learn the proper ones?