This is probably just because the coins aren’t actually fair. If the coin is slightly biased towards heads, the first throw is more likely to heads, and so are all subsequent throws. Same for tails.
That's not the problem. You can test that by using a highly secure random number generator, e.g. /dev/random in Linux, to select the initial side. Keep track of that initial side, record the side it lands on. This paper shows a same-side bias, not a heads bias.
If you take a specific coin and find that when you prepare it to be flipped showing heads up, that it is more likely to land heads up, and that when you prepare it to be flipped tails up, it is more likely to land tails up, it seems confusing to call that coin 'heads or tails biased'