I usually rate Tri-X at box speed and develop using Lauder Formula 76 (D-76 generic), 1+1, for 9.75 min, i.e., as suggested by development charts.
I read here that some photographers rate Tri-x at 200 or 250 for better shadow detail. If I do so, do I need to cut down on development time, i.e., develop for 200 or 250, not 400?
You don't need to. If you're getting negatives of the appropriate contrast with your current development time, stick to it. But if you're getting contrast that's too high, the extra exposure gives you a bit of leeway to reduce development.
Those ISOs were determined most likely when examining the characteristic curve of a film. Most higher speed films ( 100 +) are rated at the maximum rating for their curve while still retaining usable recorded density, in truth to get the best contrast and maximum workable density the exposure most be increased around 1 EV. But of course once you understand characteristic curves your exposure index should be determined by visualization.
Some developers are better at eking shadow detail from the film than others, so that will determine your exposure index to some extent as well.
I use Xtol and find I can shoot both Tmax 100 and Tmax 400 at box speed. But as the previous poster states, I sometimes underexpose or overexpose in order to comply with the lighting conditions, i.e. if I want to increase contrast in flat lighting or reduce contrast in high contrast scenes.
Those ISOs were determined most likely when examining the characteristic curve of a film. Most higher speed films ( 100 +) are rated at the maximum rating for their curve while still retaining usable recorded density, in truth to get the best contrast and maximum workable density the exposure most be increased around 1 EV. But of course once you understand characteristic curves your exposure index should be determined by visualization.