Many thanks all for your answers.
Now I'm even more confused than before.
Geoff - you're right. The trainer kite was probably LEI. It didn't have a pump.
Balugh - your suggestions are very useful. If I was buying something, I'd probably would go for Flexifoil Quark 3-Line Traine as it has a bar and I can attach it to a harness - most similar experience to proper kite surfing.
I understand why it might be worth to just to get onto a proper kite without buying a trainer kite. The problem is that a session (4h) costs me £120 - almost the same as price of trainer kite. And I don't feel I'm good enough yet to handle a proper kite (and buying all equipment necessary - we're looking at at least £1k).
On the other hand, I'm a bit worried trainer kite won't be that much of use for me. So far I had 4 lessons and I know a bit how to operate a real kite, I did some body dragging and I also had 2 (1 hour each) sessions of wake boarding.
To add to a confusion I don't think I will be doing that much of kite surfing as I live in London and my instructor will be away over winter time, and I don't want to lose acquired skills. So it's still better to go to a park over a weekend and work on kite skills.
Hmmm.. I really don't know what to do next!
Cheers,
Richard