Do you always play the way the game is supposed to be played, finish the game and move on or you do enjoy trying to break the game?

Here is a print from Into the Breach, where each mission is supposed to last 4 rounds. Your objective on the last mission is to hold up until the ticking bomb explodes all enemies and if you eventually loose the bomb, the game punishes you with more rounds (and another fresh bomb).

For those who are not familiar with this game (you should, is awsome) I covered most of the tiles with smoke so the enemies can´t attack.

Yes to the question in the title; no to the first question in the text.

(I have more pictures, but it seems the forum limits you to 5 pictures per post.)

I don't enjoy "breaking" games, yet I am all for mods with game fixes and unofficial patches (half-purist with a bit of cut content where possible).

Can't say your example can be qualified as "breaking" the game though, I've played it a bit but perhaps I'm missing something.

Here's some more.

My favorite is noparty.png
(In that screenshot, my party of 0 adventurers is fighting an enemy, and the enemy attacks. (Who does that enemy attack?))

I definitely enjoy finding the optimal route(s) or through, or character builds for a game, and finding sequence breaks in metroidvania/platform adventure games.

I don't enjoy when it's like you're abusing a glitch or oversight to completely remove the challenge of or skip a part though.

I don't really enjoy breaking games, I just enjoy being a little defiant and testing their limits to a reasonable extent. If the game tells me what I should absolutely do now, I often try doing all kinds of other stuff instead, just to see how it reacts to that and to check that I'm not missing anything because the game is trying to rush me. And I will make use of exploits if I find them and they allow me to deal with e.g. annoying bosses more conveniently.

The example in the OP sounds like creative, but perfectly legit tactics to me.

If the glitch skips a part of the game that is not fun at all, I don't mind using it. (For example, if a game has an insta-fail stealth sequence, I won't mind using a glitch to skip it if there is one.)

Another interesting case is if a glitch allows you to temporarily skip the game, but you still have to come back later. Ys: Oath in Felghana has a trick (not a glitch) that involves going to an area before you're meant to go there; you can't finish that area yet, but you can at least get enough XP to trivialize two boss fights that you temporarily skip. (The leveling curve is steep; you can't effectively level from enemies that are below your level, and a small number of levels makes a huge difference, particularly in boss fights.) (I think something similar is possible in Ys 6, but not in Ys Origin.)

Well ok, second time through that might be a good thing. But better yet would be a redone segment.

I especially like to break the systems of games that are deliberately set up to have the ai cheat.
One of the best examples of this is the rolling pin strategy in warcraft 3 playing elves.
Leverage the fact that buildings are not units for cap and systematically build and move closer keeping full unit cap only for skirmishers and heroes.
Your barracks are the rolling pin and your skirmishers are the water that help smooth out to a fine paste the otherwise in-compressible flour of anything that gets in it's way.
Once you are at their base drop two trees to constantly reinforce as units die and reap the rest of the map to support the unending wood and flesh tide.
Delicious!

Reposting my air walking screenshots from Assassin's Creed. I can't really take credit for this as it happened completely by accident.

Is not exploiting a bug or something but keep in mind the AI units (VEKS) can't move outside the smoke and they take 2 damage each round. The only exception is the Hornet at B6. If I want to win, I would need to move the F4 unit to A4 and just let the turns run out, without even fire a shot. BTW this was done in hard mode :D Yeah, this is a big problem one want to enjoy a game. Back in the day whenever I used cheats, it kill the enjoyment after a while.
I avoid those like the plague.
However, Into the Breach is a fine exemple how to do things, no matter how the enemy plays, you know the moves before you do your own, so any kind of "AI cheating" is not that important.

Most of the glitches and bugs you founded by chance or actively pursuit them? That is a fine collection.

It varies. Some I read about or see online, some I discover by accident, and some is the result of glitch hunting.

I believe a few of them are cheated, but not all of them. For example, I believe the Bard's Tale 3 empty party is one I found on my own before the days of the internet (along with an exploit in the Apple 2 version that allows you to repeatedly get a 600,000 XP reward; that was fixed in the Commodore 64 version but the empty party one wasn't), and the Oblivion one I believe just involved having an intelligence boost wear off reducing Magicka into the negatives (you can also get into this situation in Morrowind); on the other hand, I believe the Final Fantasy 5 dead party may have been the result of a little hacking (to trigger a special plot battle in place of a random one IIRC), though it's actually possible to get into that situation with a couple of glitches I learned about through the internet (use Teleport in the Ancient Library to wrong warp, learn Quick, and then have a character cast Quick followed by Self-Destruct (Explodor); if that kills the last enemy and the last party, you're awarded the victory).

By the way, here is the video (not by me) that taught me that Final Fantasy 5 glitch (includes a variation of the ending that's normally impossible, so *spoilers*):
https://www.youtube.com/watch?v=YhMl87C8bhA

No I don't think I've ever had an interest in that, but I do sometimes try to perfectly understand game mechanics. Most recently, in Caesar 3 when I wondered what the algorithm is for a market lady / house interaction.