Even reddit's post text editor has BOLD and italic. Most other systems seem to also have underline, with strikethrough becoming more and more common.
I'm just imagining all the other ways we could transform text, but these seem to be the only universal 2. Why is that? When did they (and maybe underline too) become standard?
Differentiating between normal and italic print has its root in the practices of book culture around 1500. Normally, latin text was printed in an italic typeface, which we call Antiqua, the serif one with slabs and so on, which would later evolve to Times New Roman. The Italian Aldus Manutius came up with it, he wanted to imitate the time's handwriting (people really had beautiful handwriting for official documents back then). Fun fact: Many of the most beloved serifs actually stem from this time period! Typefaces like Bembo, Garamond are well past their 400th birthday and still trucking.
Let's discuss the use of non-italic type. In Germany for example, text of the German language was printed in blackletter: the weirdly gothic looking type you see when watching Illuminati. Differentiating between these two types was simply common practice.
However, the key part was when a book was featuring both languages, e.g. German and some latin expressions here and there.
Investigate by yourself: What was the solution for those texts?
What the typesetter did was using blackletters throughout the German text, and switch to italics when a Latin word showed up. From then on, the use of blackletters swindled over time, but italics were to stay.
What then happened was more of a genesis of our todays typefaces, were italics continued to be an option to mark certain words.
This evolution of type is better portrayed by a Historian of Modernity - there are a whole lot of aspects in the mix that led from blackletters to our minimalitic, rounded sans serifs and "grotesques". In a nutshell however, people realized the power of advertising publicly. They needed typefaces for that role and the old ones - e.g. Bodoni and Didot, think of the VOGUE Magazine's logo - didnt cut that. New ones had to be specifically made for that role, which resulted in a cleaner look. With that progression, heavier "weights" as we call them will likely have found their way to the printing world as well: Bold, Semibold, Black, etc. were not needed in earlier times, especially because even having one font of a typeface was very expensive.
Now, there is one method of marking words you didnt mention which has been around for quite a while as well, and that is coloring. Already in the 1500s, red word coloring was sparely used, foremost in the title page to mark someone's name. This didnt enter the main body of the books, since it required a doubled workload in the printing section. But it was already used nontheless. When techniques improved in the 1800s, it was used more often.
I hope I could shed some light onto it.
Sources (German, sorry):
Kapr, Albert: Fraktur. Form und Geschichte der gebrochenen Schriften 1993.
Heyden-Rynsch, Verena von der: Aldo Manuzio. Vom Drucken und Verbreiten schöner Bücher. 2014