Transliteration differences

Romanization > specifically into the Latin alphabet.
Transliteration > any other alphabet (Cyrillic, Greek, Devanagari, Hangul, Farsi, etc.).

That's just about it.

As for thread searching, there's no such option at the moment.

Isn't romanization more complex than "transliteration to Latin alphabet"?
If I understood it correctly, transliteration is a 1:1 letter/symbol-wise copy whereas romanization respects pronunciation and thus can transcribe the same source letter with different target letters.

So Ukrainian Володи́мир would be transliterated "Wolodimir" but romanized "Wladimir".

Or as a ficticious example, if the source language noun "§o§a" were pronounced "ksoh-shuh", a transliteration would e.g. render it "xoxa" but a romanization would use "xosha" in English and "Xoscha" in German.

For japanese is katakanization. Writing the words in the katakana syllabary

The examples you provided have nothing to do with transliteration or romanization. At the most they could be considered as colloquialisms like "ain't", "gonna", "gotta", "kinda", "wanna", "wanta", "kinda", "hafta", "tryna" etc. but transcribing “bur I” instead of “but I”, “dient” instead of “didn't“, “shouent” instead of ”shouldn't” would be wrong and confusing.
A transliteration is taking a piece written using an alphabet and making it readable in another (naturally one still wouldn’t know its meaning).

https://en.wikipedia.org/wiki/Transliteration
https://en.wikipedia.org/wiki/Romanization

It depends: some colloquialisms are significant, others are not. Only the first ones should be transcribed. But no one would probably be able to set up general rules to determine what is significant or not.
Usually, lyrics are copyrighted with a certain text: it should be stuck to it.

It is very difficult to determine which colloquialisms are significant and whether or not to transcribe them in a language you don’t speak fluently. Native speakers usually just “know” based on experience of what “sounds right” (although what is “right” can be very subjective). Learners don’t get that sense until they are near fluent, and maybe not even then.

Not that you shouldn’t try—but definitely be open to correction and critique from the native speakers!

The Ukrainian “Володи́мир” cannot be transliterated as "Wladimir", because it is pronounced exactly like “Volodymyr” in the Ukrainian language («Володы́мыр» in Russian alphabet). Cyrillic letters representing vowels are pronounced differently in Russian and Ukrainian. Cyrillic-based national alphabets are not the same for different Slavic languages! Also, many Latin letters indicate different sounds in various European languages. Compare, for example, the sounds indicated by the letter “j” in English and in Spanish. Many consonants of Slavic languages can be written only by a combination of Latin letters. In transliteration to Latin, combinations of letters typical of the English language are usually used. So romanization or englishization?
I don’t see much sense in transliteration at all. It is easier to learn the national alphabet than to understand what transliteration rules the author of transliteration was guided by.

If I am right, UNESCO has published rules for the translitteration of non-latin written languages in international institutions, and elsewhere.
Some may be different from what we are used to do in common usage. For instance, in Russian, they use "shch" where French-speaking people write "chtch".
However my opinion is that for the sake of uniformity we should comply with these rules.
In any case, the worst solution would be to use diacritics. Leave that to linguists and philologists (among whom I am!). Or else the only solution would be to use AIP/IPA's "alphabet".

What are we speaking about, exactly? Is it "translitteration" or "transcription"?
In the first case, it is only a matter of CONVENTIONAL correspondence between alphabets, no matter how the letters ar pronounced.
For instance, Greek "delta" is normally translitterated as "D", athough it is pronounced "th" (the voiced variant, as in "this').

Just think of the problems the librarians have to face to arrange authors' names in alphabetical order!

If I see “Володимир” I would have no idea of the way it is pronounced. If I see a transliteration of it as “Volodimir” then I would have no problem following the recording of the song and possibly sing along with it.
I still would have no idea of what it’s saying but that is not the point of a transliteration. It’s merely to give an idea of the pronunciation of a piece that was originally written using an alphabet that one is not familiar with. The precise pronunciation can be discerned by listening to the recording and it really doesn’t matter if, in English, “Vladimir” would be more appropriate.

In fact, you will have a problem, because (as I wrote three times here) in Ukrainian Володимир is pronounced as Volodymyr, and not as Vladimir (Wladimir) or Volodimir. The letter "и" in the Ukrainian language denotes a sound different from that which it denotes in Russian. And therefore, it cannot be transliterated and must not be transliterated from Ukrainian into Latin with letter “i” as the automatic system of romanization on LyricsTranslate does.

Above, I provided a link to an official document, fixing at the legislative level the rules of transliteration from the Ukrainian language into the Latin alphabet.

Most of them! Have you ever heard of things like the Library of Congress, the British Library, and so many others? As a professional, I have never heard of libraries sorting their books on the shelves according to the languages -- except for ancient manuscripts.
Moreover, the problem is not with shelves, but with catalogues -- either on paper or digitized.

Of course, the "alphabetical" approach may seem very formal, and irrelevant to those whose concern is only pronounciation.
Now, if I may enter in more personal considerations: I am a specialist of the Breton language (Brezhoneg), an endangered Celtic language spoken in the far-west part of France (Brittany). Although it is undoubtedly one single language, it is split into an infinity of dialects, with major differences in pronounciation, often at a distance of less than 30 km. (The grammar remains the same altogether.) Intense work have been displayed since the beginnings of the 19th cent. to set up an orthographic system which could allow to transcribe all words in the the same way, independently of their local pronounciation. Sticking to the oral forms would be a complete disaster, and the final wreck of this language within a few decades. (A process of the same kind is currently on its way for the Rheto-Romanic dialects in Switzerland.)
An what to do with the ancient, hieroglyphic Egyptian language, which can be dealt with by specialists who don't know exactly how it was pronounced?
What, again, with Latin, which is pronounced in so many different ways according to where you have studied it?

I see this to a lesser extent with my local (Cologne) dialect ("Kölsch"). Since 1983, the "Akademie för uns Kölsche Sproch" has been compiling a dictionary that tries to transcribe the spoken dialect. There we also see a good example for the difference between transliteration and romanization: The dialect word for "gut" ("good") is written "god" (which would be pronounced like "gohd" but with a short "o") but is pronounced "jood" (or, phonetically for an English speaker, "yohd").