Is There Different Ways to Write Katakana?

So, I have been trying to learn how to write in Hiragana and Katakana recently, so I have been carrying my "Kana de Manga" around with me with a little notepad and jotting down a new letter or practicing an old one whenever I have the time to do so. However, I have noticed something curious recently, it seems like the Katakana characters in my book are similar, but not always the same as the ones I see written online. The ones in my book are very rigid, with straight easy to follow lines, whereas the Hiragana characters are kind of more loose and flow-y. Up til now I thought that was a difference between the two. However, now I am seeing Katakana characters that are more curvy and connected a lot more often than what I have seen in writing thus far. If I write them the way my book presents them to me, am I writing them wrong? Should I write them more lax and connect the brush strokes, with less emphasis on straight lines and definitive parts? Thanks in advance for all the answers, I just don't want to be writing them wrong.

Think about the differences in English between typography, block letters printed with a pencil, writing in cursive, and writing in calligraphy.

None of those are right or wrong, they are just used in different ways and in different situations, and individual letters can have different appearances; quite different in some cases. Think of the lower case "g" in the Times New Roman font...does anyone write a "g" that way when they are jotting a note to themselves?

Also, Japanese has different fonts, just like any other language, and the individual strokes can look quite different between different fonts.

You referred to "brush strokes"; do you mean literally writing with a brush and ink? That is going to look totally different from how kana look online, in printed material or in someone's normal handwriting done with a pen or pencil.

So, long answer short, no, you are most likely not writing them wrong if you are writing them in accordance with whatever textbook you are using.

Just for your reference, historically Katakana is called as man`s character and Hiragana is called as woman`s character.
You should write strongly Katakana with lines and write Hiragana with curved and soft touches.