Is prion a term used to describe the normal form of the protein as well as the disease causing form?
The normal isoform of the protein is called PrPC, which stands for cellular prion protein, while the infectious isoform is called PrPSC, which stands for scrapie prion protein.

According to Riesner (2003):


  The biochemical properties of the prion protein which is the major, if not only, component of the prion are outlined in detail. PrP is a host-encoded protein which exists as PrPC (cellular) in the non-infected host, and as PrPSc (scrapie) as the major component of the scrapie infectious agent. (emphasis mine)


If you search for "cellular prion protein" you're gonna find several papers that use the name prion protein to the normal isoform. Some examples:


Prado, M., Alves-Silva, J., Magalhães, A., Prado, V., Linden, R., Martins, V. and Brentani, R. (2004). PrPc on the road: trafficking of the cellular prion protein. Journal of Neurochemistry, 88(4), pp.769-781. 
Ramljak, S. (2008). Physiological function of the cellular prion protein (PrPc_1hnc). 1st ed. Berlin: Logos-Verl.
Pantera, B., Bini, C., Cirri, P., Paoli, P., Camici, G., Manao, G. and Caselli, A. (2009). PrPc activation induces neurite outgrowth and differentiation in PC12 cells: role for caveolin-1 in the signal transduction pathway. Journal of Neurochemistry, 110(1), pp.194-207.
Martins, V., Mercadante, A., Cabral, A., Freitas, A. and Castro, R. (2017). Insights into the physiological function of cellular prion protein.


And many others.

Therefore, following this nomenclature, the answer to your question ("Would I be correct in saying 'The prion protein is normally involved in synaptic transmission but can turn into a disease causing form'?") is yes. The difference is the adjective: cellular or scrapie.

Finally, pay attention to this: you have two different questions here. In the title you say "Is prion a term used...", but in the last paragraph you say ""Is the prion protein normally involved in...". As extensively discussed in the other answer, the term prion alone (instead of prion protein) is normally used only when referring to the abnormal isoform. More on that here: https://www.cdc.gov/prions/pdfs/public-health-impact.pdf

Source: Detlev Riesner; Biochemistry and structure of PrPC and PrPSc. Br Med Bull 2003; 66 (1): 21-33.