Are these shoots from my blackberry apache?

I have suckers coming up in my lawn but I'm not sure what they are. 2 or 3 years ago I planted a blackberry apache. I asked at the garden centre if it was invasive and they said it wasn't, though once established it can be quite vigorous, so being a thornless variety I went ahead and bought it.
I now have shoots coming up in my lawn, but I don't know if they are all the same thing or not. Some shoots appear to be thornless and others elsewhere in the garden and grass have thorns - lots of them, but they are quite soft. I've attached some pictures and I hope someone can help.
I'm not sure if the ones with thorns are from the same thing as the leaves are bright green and the others are quite red. My suspicions are that the red ones are from the blackberry and the green are from something else (could they be rose suckers? I don't have roses so not sure what they look like, for all I know next door could have them?). The green thorny ones are popping up in my garden in a few places and I just keep glyphosating them but I'm starting to get concerned now that they are appearing in my lawn!
If anyone can identify them I'd be grateful and any tips on what to do about them. If any of them are blackberry shoots I'll be annoyed because I asked at the garden centre before buying it. If that's what it is I'd rather not have the blackberry plant and kill the whole thing off.
This pic shows my blackberry plant and you can just see the red shoots in the lawn:

This is a close up of the red shoots:

You can just about make out where the shoots are in my lawn:

These are the other shoots which I'm pretty sure are not the same - they are about 3 or 4 yards away from the other shoots, so not particularly near to the blackberry:


This is one of the same weeds that's just come up under my pieris - I thought this would be a good pic because you can see the leaf and thorny stem quite clearly.

Any help would be appreciated.
Thanks!
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Do you have any Raspberry canes nearby?
not in my garden. No idea what is in next door's garden.
We have a thornless blackberry which throws up the most appallingly thorny shoots every now and then, the leaves on them are different to the standard ones too.
do they dig up fairly easily - it's chucking it down with rain at the moment so it will have to be done one evening this week, or next weekend. Some are too near to my plants to be dug up I would have thought - they've come up immediately next to the stem of the other plant in some cases.
Do they normally send out lots of these suckers, or is it just every now and then? If they send out as many as have appeared in the last week I'll seriously have to think about whether I want to keep the blackberry at all.
So do you definitely think they are both from the blackberry and not wild raspberries now then?