Yes, you pretty much have the procedure right but you would add the "Cinelog-C_to_Alexa_Log-C_EI800_with_Film_Matrix" or "Cinelog-C_to_Cinelog-C_Film_Matrix_RGB" before your color correction i.e. the node chain to add to your imported ProRes log masters would be:
1. Cinelog-C_to_Alexa_Log-C_EI800_with_Film_Matrix or Cinelog-C_to_Cinelog-C_Film_Matrix_RGB (this transforms Cinelog-C to Alexa Log-C with either Alexa Wide gamut or Alexa Film Matrix primaries)
2. node(s) for color correction/grading - (leave untouched for now)
3. Film Convert - select Alexa Log-C or Alexa Log-C Film Matrix (depending on which profile you used in step 1 above)
4. Use the Film Convert curve control to adjust contrast and sat control for saturation
5. make changes to color using the node(s) added in step 2 or just use the Film Convert controls
6. check levels on scopes and export your graded video.
Alternatively, select any other Cinelog-C to '
other colorspace' lut and use the relevant Film Convert input profile. i.e. Cinelog-C to sRGB lut and Film Convert's default sRGB input profile.
Film Convert is emulating what the image will look like if it were to be printed to film using an ARRI laser
http://www.arri.com/archive_technologies/arrilaser/ or similar. The one step I am not sure about with Film Convert is what (if any) Color Print Stock or color reversal film they emulate!?
In a real film workflow the image would be captured on film negative, the negative would be developed and this would then be printed to a color reversal film, color timed (i.e. a primary grade using printer lights) and printed to a color print film master (i.e. Kodak Vision 2383 etc). This is then duplicated for distribution.
If Film Convert does not include the last step (i.e. the Color Print Film) you will need to add a good PFE lut (Print Film Emulation) and set Film Convert's Film Curve to zero (i.e. so you only get the color and not the contrast).
Depending on the Film stock emulation selected and print emulation lut you may get very thick or sometimes out-of-gamut color that will require color correction in nodes before the Print Emulation lut.
This is exactly the reason PFE luts exist i.e. so you can avoid any nasty surprises when your digitally captured video is printed to real film - it allows you to see color errors and correct them before committing to expensive printing.
The Cinelog-C Film Look luts do everything in a single lut

- we have added corrections to adjust for possible color errors, added saturation and other color tweaks for the 'look' and use the gamma/tone curve of the print stock for contrast - but because we added saturation and color tweaks/fixes we don't call them PFE luts.
re: your last question: yes, Cinelog-C and REC709 are very different. Cinelog-C is a scene referred colorspace and REC709 is a display referred colorspace. The Cinelog-C to Cinelog REC709 (legal and full range) luts are based on the REC709 transfer function and all Cinelog-C to rec709 luts transform from Cinelog-C primaries to REC709/sRGB primaries AND the include our s-curve to fit the DR of the image into a REC709 display space.
The Cinelog-C to REC709 lut is the same but does not have an s-curve (i.e. it's just the rec709 transfer function) and highlights will clip. The rec709 transfer function will allow for highlights above 1.0 (floating point) up to ~2.2 (on a scale of 0.0 - 1.0) but any values above 1 will be clipped when rendered - this is what an s-curve is for

Get well soon!!
