I'm no expert but as someone who has done reading up on the same subject (DIY'er with a condo built in the 70's), that lino does not strike me as having asbestos in it.
Even if it did, you are working with the "safest" form of it. The really dangerous stuff is the kind in old plaster, ceiling tiles or vermiculite in attics.
You can buy a full blown dust mask with filter designed especially for asbestos. Based on those pics, it appears to be coming up pretty clean. Cut with a sharp knife, bag and get it out of your house you will probably be just fine. This one job isn't likely to have any impact on you at all.
I totally understand your concern though.
My girlfriends dad told me stories not long ago how they used to cut sheets of asbestos with a table saw....good Lord. He was recently diagnosed with COPD but seems to be coping just fine.
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