Hello Lucy

I am dying to know where in Dorset you are, not least because we have relatives in SW Devon and love to find some Red Rubies to admire when we visit...

However, to address your question, from a beef/sheep farmer's perspective, haylage is silage that was got a little bit drier. There's no hard and fast distinction.
Years ago, farmers put additives in silage to make it ferment quickly, but that is generally not done these days. I believe this is because the modern techniques for baling and wrapping do away with the need for the ultra-rapid fermentation - but if that's not the reason, someone will be along shortly to correct me and give us both the real reason!
Anyways, haylage and silage are essentially the same thing, but generally it will only get called 'haylage' if it's pretty dry. If the 3-5 days it took from cutting to baling were dry days, then it's certainly haylage.
I may have the wrong end of the stick, but if your concern is feeding it to the horses, there's a thread in Horses about feeding silage; if you read it you'll see that quite a few folk do feed silage to horses. And quite a few wouldn't, so it may not help, I don't know!
Welcome to the forum, we will all be very interested in how you get to grips with running your new farm and be glad to help out where we can.
Cheers, Sally