"Tack" from "Tackle"?

Got a question regarding terminology:

My neighbor adamently disagrees with me that the word "tack", referring to horse
equipment and tools, comes from the longer term "tackle". I've known this since
I rode as a child, when my father took me to tackle shops, or we put our saddles
up in the tack room.
The words are used alternately - am I correct?

And I answered back:

Sorry, but I'd have to say your neighbor is correct-- it looks possible that 'tack' was derived from 'tackle', but today horse people refer to their equiptment as 'tack'. I did a Google search and found very few relevant results for "horse tackle"-- just a handful and mostly by non-horse websites.

I don't know what part of the world you are in, but in some places the terminology is different-- for example, a 'Grulla' colored horse here in the US might be a 'blue dun' in the UK. However, if "tackle" was used in other parts of the world, I'd expect to see it more widely used online. It is possibly that 'tackle' as used more often in the past and has been shortened to 'tack' in more recent years.

I looked up "tack" in the American Heritage Dictionary online, and it says "short for tackle" under the horse definition.

If anyone has any suggestions or additional info on this, feel free to comment.