Can I keep a limit break for later?

I have filled up my limit break. How exciting! Flashy colors! I can now deal serious damage! Wow!

...but I'd rather not use it on that enemy that would go down regardless. In particular the menu screen speaks about multiple "limit break levels," so it seems that it should be possible to save those up for boss fights and rainy days.

What can I do to not use up my limit now?

As others have noted, when at limit levels 1, 2, and 3 using the initial limit break a certain number of times unlocks a second limit break at that level. At limit level 1, using your limit skill 8 times unlocks the second limit at that level, level 2 is 7 uses of the limit skill, and level 3 is 6 uses of the limit skill. It is advantageous to use limits as many times as possible early on in the game to unlock the second limit skill at each level. This allows you to learn new limits as you level so when you get the character's level 4 limit manual, you can use it right away instead of grinding out limits (all previous limits must be learned before you can learn the level 4 limit from the manual).

Judging by your screenshot, it appears you are very early on in the game, so I would use Braver and Big Shot as often as possible in order to unlock Cross-Slash and Mind Blow. As you progress in the game, you will find command materia that will help you avoid using limits/magic/defend and still contribute damage (e.g. Mug, Deathblow, Throw) if you are needing to save your limit break for another fight. Enemy Skill is also a powerful command materia you can use to avoid using limits if necessary, but like magic, Enemy Skill commands use MP.

No, you cannot save your Limit Breaks unless you are prepared to only fight with magic or items for the duration. In both cases, you end up using a finite resource (MP or items) while also stalling the limit meter recharge, so its not very useful in general.

Note that many characters need to keep using their limit breaks for each level to unlock an alternate one (e.g. Braver / Cross-Slash). You will be better off to use limit breaks asap to progress in the unlocking and have more options.

If you know there is a boss battle coming up then it may be worth fighting with magic or items for a short while (so that you start boss fight with multiple charged limit breaks). Keep in mind that the level 1 breaks charge up really fast, while level 3 or 4 is far slower - you don't want to stall on level 1 limits when they will be up in a short time anyway.

Re. the menu showing multiple levels for each limit break - these cannot be charged concurrently, you have to select one and then charge it from empty (e.g. for Cloud, you can't charge Meteorain, switch to Cross-Slash for random battles, and then switch back before boss fight expecting meter to be full).

You can defend and let Barret finish the enemy, you can kill it with magic, you can run, or you can kill the enemy with an item. You cannot use the ATTACK command with Cloud.