You've just answered your own question.
Transits are normally commercial vehicles, a lot of dodgy traders try to cut costs by not paying for the disposal of commercially produced waste.
Hence as council run tips are normally for residential waste you have limits on certain things, such as the type of vehicle ('commercial' ones are normally banned or require a permit with the registration on it etc), and size of trailers.
By the sounds of it your council allows them, but the chances are if you were there on a regular basis the tip staff would note the reg and ask for your CT bill to help rule out you really being a dodgy waste disposal guy/builder etc.
The exact limits depend on the local authority, for example our one requires a free permit to dump tyres, or things like fridges (you're normally allowed something like 2 fridges or freezers, and 2 or 3 tyres a year).
I've never had an issue with our tip even when we've made multiple trips in an afternoon* - I suspect the use of a normal car, and the fact we're often taking in green waste, or quite wildly assorted stuff from tidying up or doing DIY helps.
On the other hand I know a local decorator who managed to get his entire family banned from the tip, they didn't realise the people at the tip were able to remember the same people/cars turning up on a regular basis with multiple trade tins of paint and obvious decorating rubbish.
*I think our record is about 4 trips in an afternoon, with about 8 over the week, we'd accumulated loads of waste from tidying up an overgrown part of the garden, and masses of hedge trimmings from cutting about 12-18 inches off our hedge (as well as stuff from the loft and garage). The "ton" bags builders sand/gravel come in are great for taking garden waste to the tip if there are two of you, and a reasonably big car (just be careful to make sure you can lift the bag, grass clippings are heavier than you'd think
).
)
