- Proprietary - No access beyond remote endpoints
- Downloadable - You can run it, but there are restrictions and training data/code isn't public and/or under FOSS license, nor are the weights under a FOSS license
- Open weights - The weights are under a FOSS license and downloadable without restrictions, but not all of training code/data is public or under a FOSS license
- Open source - The model architecture, weights, training code and data are all under a FOSS license, and you can freely download and use them without any restrictions
Few models hit the mark for "open source model", many models people call "open weights" would go under "downloadable" here, which personally I think would be accurate.