Would either of those types of tickets have sufficed for the TLK train?
No - you need to have a TLK ticket.
And if not, why have an automated machine that sells tickets for only two types of trains?
Different companies. Unfortunately - Poland didn't introduce "impartial retailing" laws for the railways.
And a more-general request - can someone in the know outline the differences between all the train choices in Poland? I've lived here for 3 years, yet I am pretty much in the dark. Thanks for any and all constructive input...
Right now, you've got several different types - but I'll try and do my best.
EC - EuroCity - international trains operated by PKP InterCity.
EIC - Ekpress Intercity - the most expensive. Usually very nice trains, modern, fast, etc. The most expensive (belongs to PKP InterCity)
TLK - Twoje Linie Kolejowe - the "cheaper" PKP IC brand. The quality can be utterly variable - sometimes nice, sometimes not so.
InterRegio/RegioEkpress - the long distance brand of Przewozy Regionalne. Again - quality can be utterly variable. Usually about the same price or a bit cheaper than TLK.
Regio (also known as osobowy) - the local train brand of Przewozy Regionalne. Cheapest tickets.
They're the nationwide operators - there are some local ones, priced at the same level as Regio tickets.
The important thing is that you need to make sure that your ticket is valid for the train that you want to travel on. However - sometimes - if a train is actually cancelled, then you can hop on a different service operated by the same company.