I was thinking, are there any softmods available for the PS1 or PSOne to play backups?

Please let me know, thanks.

Nope, swap trick and chip are your only options.

I found some videos on YouTube of people claiming that they softmodded a PSONE.

A result in using Google. I do think that the only methods are basically what LJTF stated above.

How is this "swap trick" done? Does it work on both the PS1 and PSOne?

I don't know how the PSOne's work with regards to the lid switch etc, but I'd imagine you would do the same thing.
It's the same as on the Saturn. You tape the lid switch down (Or put something in there to hold it down) and load a retail game. At a precise point (You can HEAR the drive motor spin down) just before the SCEA logo shows, you quickly grab the retail disc, and exchange it with your backup.

It's been years since I've done it, it's more involved than that, you require Google.

It is more or less that simple. It is a pain to do, but not that hard. My PS1's laser can barely read a pressed disc though so backups are out of the question.

GC- Why not just go with a Slim PSP with TV-Out? It is essentially the same thing, perfect emulation and all.