Definitely not the same. For me, word processor is a computer software program, such as WordPerfect and Word. A typing machine would probably be a typewriter, as RoadFox suggests, although I don't think I've ever seen the phrase used.
Though now word-processor has the meaning above, there was a time in the 1970's when this term referred to a machine that looked like a typewriter, but would allow you to compose and revise a small amount of text in a window above the keyboard. Then you would finalize it as text to be printed. PICTURE.
A typewriter prints directly on the page as you type, without the intermediate step of seeing the text in the window. AN ELECTRIC TYPEWRITER.
Back in the early days of computing, some companies, e.g. Wang and Digital Equipment Corp., sold machines dedicated to producing documents. These were called "word processors" to distinguish them from typewriters, which were slowly advancing in capability.
Dedicated world processors soon disappeared in favor of general-use computers using software (not the redundant "software program" - sorry, it's a pet peeve) that not only handled the processing of text, but advanced layout as well.
"Typing machine" is not common English, but resembles non-idiomatic translation from some other languages.
It might be worth pointing out that the very few electronic machines still manufactured solely for the purpose of creating text documents are still called "word processors." (See this one for example.) Because I am abnormally obsessive about semantics, I still refer to MS Word and Wordperfect as word processing programs. But I don't expect anyone else to do the same.