They're both lossy, just in different directions.
Now I'm beginning to wonder if you really haven't gotten yourself confused.
for a given number of output bits.
Forget the output bits. It's the input bits that matter. PNG doesn't truncate¹ the input data. If you want to do that in order to achieve the same output size, that's your business, not your file format's.
There just aren't two ways about it. PNG isn't lossy as long as you define the term "lossy" in a manner everyone else who talks about compression will understand. Using Information Theory as an excuse to suggest we hold output size constant and then redefine "lossy" to mean "lower compression ratio" doesn't help understanding or foster meaningful communication. Nor is it cool.
1. I'm using the term "truncate" loosely, of course. I mean to say that conversion to PNG doesn't delete pixel data from the source image.
"My two cents aren't worth a dime.";