BadGoldEagle - I was going to mention what you stated but realized that the OS in question is 8.1 and in 8.1 the format is in HFS+. To install System 7 one needs to format the drive in HFS. This would be a total loss in data.
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68K macs can't boot from an hfs+ formatted drive. So the OP's mac is not formatted hfs+.
I'm almost tempted to say "Wanna bet?" But my LC 475 is on the frtiz and my only other 040 machine is a IIci with the DayStar 040 Turbo in it. It has a 2.1GB Drive and it is formatted in HSF+ running OS 8.0
OS 8.0 can run on HFS for partitions 2GB and under and HFS+ with partitions 2GB and over. OS8.1 can not run under HFS without difficulties, and 8.6 can not run on HFS at all regardless of partition sizes.
Pushing my experience further, several schools I worked for had Performa/LC 475s, 575s & 580s running OS 8.1 on HFS+. When one puts in a 575 or 580 recovery/install CD with System 7.5 or 7.6 and you click on the hard drive, a window pops ups with one folder in it and a TeachText file with the file name "Why cant you see all your files?" which explains why. When that happens when you boot from System 7, the drive is formatted in HFS+ and not HFS. In fact motley2659 should boot in System 7 and see if he can see all the files or not. If he can not, then he needs to reformat the drive in HFS.
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motley2659 - Which system 7 you set up is on you, I'm not questioning that or the reasons behind it. You should verify what logic board in your 631CD and many of them were upgraded to a 6312CD (PowerPC) with a logic board swap. That's simple to do by sliding out the logic board from the rear and look at the CPU. If there is a square "pie pan" heatsink on the CPU clipped onto the logic board, its PowerPC; if it has a "castle top" heatsink glued to the cpu, then its a 040. If there is no heatsink, you can read it off the chip.
Then and only then when you find out what you have, you can use a 68K/PowerPC FAT Stripper/Checker and strip out what code you do not need. If you have an 040, you do not need PowerPC Code in your system and Apps. This will shrink your programs/system make things load faster, use less memory and adding all these benefits together - run faster.
And note: the 631CD came with a 68LC040 - a 040 without a FPU. If you can swap out the FPU-less 040 with a full 040 with a FPU, things will run faster as well, without touching the oscillator! At 33MHz, the 631CD is a kick ass machine with a full 040, only a few 68K machines can touch it performance wise.
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Back to topic - In seeing and reviewing this, the only option there is, is format the drive in System 7 to install it. if there is a need to have both Systems/OSes on the drive, then one needs to create two partitions and makes one for System 7 and OS 8... and things are going to get messy from there.