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Yes, [free] androgens, including testosterone, tend to lower SHBG. In turn, lower SHBG leads to lower total testosterone, without much effect on free levels.

The second reference is just plain wrong.

If our SHBG levels are higher, we’ll have less testosterone or oestrogen available. If our SHBG levels are lower, we’ll have more testosterone and oestrogen available.

This reasoning depends on total testosterone being held constant. As I have explained many times before, this is not physiological. The HPTA does not regulate according to total testosterone; it sets free testosterone. For those on TRT a fixed dose rate leads to constant free testosterone, not constant total testosterone.