The entire verse (in Hebrew) reads as follows:
לא יהיה כלי גבר על אשה ולא ילבש גבר שׁמלת אשה כי תועבת יהוה אלהיך כל עשׁה אלה ף
From right-to-left, these words are:
לא - H3808 - Not (when paired with a verb, considered imperative - "Must not")
יהיה - H1961 - Be, become, come to pass, happen
כלי - H3627 - Article, implement, vessel, object; can include concepts of hunting, war, music, tools, or equipment
גבר - H1397 - Man (generally), strong or valiant man, warrior - from H1396, meaning "strong" or "great"
על - H5921 - On, upon, over
אשה - H802 - Woman, wife, female
ולא - H3808 - Not (repeated from above)
ילבש - H3847 - Wear, put on, be clothed with
גבר - H1397 - Man (repeated from above)
שׁמלת - H8071 - Clothing, mantle, garment
אשה - H802 - Woman, wife, female
כי - H3588 - Because
תועבת - H8441 - Disgusting, abomination, unclean, wicked
יהוה - H3069 - Jehovah
אלהיך - H430 - God
כל - H3605 - All, everything
עשׁה - H6213 - To do, fashion, make, act, effect, bring about
אלה - H428 - These
As noted in the article you read, the wording is actually different between these two categories. The prohibition for women does appear to be focused upon the tools or instruments of the man, not his clothing.
Based on these meanings, we could state the verse something like this:
It must not happen that the [gear, equipment] of a man* are on (worn by) a woman, nor should a man* put on the clothing of a woman. God is disgusted by both of these.
To answer your specific question, it comes down to how one translates גבר - "man" or "warrior". Note that the word most frequently translated "man" in the OT is מאיש - H376. This is also the base word for the word translated "woman" here. On the other hand, the word גבר is overwhelmingly translated "man" rather than "warrior".
If it were intended to be a blanket statement against women wearing men's clothing, he probably would have written שׁמלת מאיש instead. For the second part, it does seem slightly broader in context ("clothing" vs. "equipment") but the object is the same "man" as before.