It's the kind of question that I struggled with for a good while.
Bought double fold bias tape usually come with two folded edges which are each less than half the width of the main body - like this:
Inastri 25/5/ 5 mm Acetate Satin Bias Binding, Red 2020
https://www.amazon.co.uk/dp/B00J5BPQDQ/ref=cm_sw_r_cp_tau_bjk1EbSW08E9PSometimes though the folded edges come almost to the centre though and this is what you will get if you use your tape maker.
In theory one edge is shorter than the other - this is to take account of the width of the fabric edge it is folded over and as Lachica says, to ensure the tape is caught on the wrong side when stitching - but sometimes to me they look so similar I can't tell the difference. You're correct in that your tape maker will make the two fold over edges the same width - this is single fold bias tape. But if the fabric edge you are binding is narrow, not bulky I don't think that will matter.
And yes, your 1/2" tape maker will make 1/4" double fold tape when folded in two.
I hope this helps. It confused me for a long time but I've used double fold tape now on a number of projects so I think I have it straight!