Wheel & lug nut question

I just acquired my Tacoma, a '13 DCLB base model. The only "mod" by the PO (to my knowledge) are the wheels/tires. I'd like to replace some of the rusty lug nuts and discovered that the nuts that are on it are apparently different from the stock lug nuts.

I believe these wheels are TRD Off-Road take-offs? (Showing my ignorance here...)



The truck currently has nuts like these:



But Amazon (I know, reliable source) says they won't fit my Tacoma (though obviously, they do.) Instead, it points to a more conventional style:



So, am I correct in assuming that the conventional nuts (bottom image) are stock, whereas the nuts with washers are required for the TRD wheels?

If so, are the TRD wheels wider? Different offset?

I'll be mounting snow tires in the winter and also trying to figure out if I should go with these rims or get dedicated snow rims to fit 245/75s vs these 265/70s.

 

The top lug is for oem alloy wheels, the bottom lugs are for oem steel wheels. However, the top lug is also designed for the oem steel spare, but as a temp measure as the contact area is smaller than with a regular acorn lug.

 

The lug nuts you have pictured in the top are for the alloy wheels that you have on your truck. These are the shank style with the washer. The tapered lug nut you have pictured on the bottom are for steel wheels and cannot be used on the alloy wheels. If you need some info about the lugnuts here is a link to a thread with more info:

https://www.tacomaworld.com/threads/parts-help-needed.809058/#post-29020158

 

Thanks all for the info. Much appreciated!