It is impossible to show that Putin is part of a "liberal democratic" government.

His state is a multi-party parliamentary democracy with regular elections and liberal instutions such as a constitution, separation of powers etc. etc. It is, without a doubt, a liberal democracy.

You are correct that he heads a 'regime' which is the term for an authoritarian government.

I also use the term "American regime". Although, yes, I suppose that in the end it is an authoritarian regime too, as are all other liberal governments.

He heads a fascist regime, as defined by everyone (notwithstanding DAR users, perhaps).

So apparently, fascism is when you kill fascists, imprison fascists and are disliked by fascists. Naturally, liberals are more correct when they say that anything they don't like, from Hitler to Pinochet to cheeto burito man, is what fascism really is.

This is the first I've seen the phrase "liberal democratic regime" for anything (as it's a contradiction in terms).

It is not at all, you are projecting your preconceived notions about specifics words and phrases onto the world at large, hence your confusion.

An the term does not apply ot Putin (who would also admit that his regime is absolutely not liberal).

That's because he's ignorant or a liar. Zinoviev already called this out long ago.

He wants us to think it's democratic, but when you remove all opposition before an election, you don't have a democracy.

All Western governments do this too so apparently you are wrong.

What some fascists think of Putin is beside the point (not all fascists think alike; in fact there is a wide variety of them around the globe).

Perhaps you should ask Russian fascists what they think of him.

What Putin is doing is literally fascist, by all definitions of the term.

No, it isn't at all, and you are a complete clown for thinking otherwise. Military action is not exclusive to fascism and does not make a regime fascist, otherwise every single state in the world would be "fascist", starting from the dawn of time.