Hi
By all means you could start woodturning for comparatively little expenditure but if you decide to continue with the hobby and expand your capabilities you will almost certainly replace all of your 'cheap' purchases with those of higher quality - ergo, no longer the cheapest route to go.
I very much doubt that your desire to turn small bowls will not expand to want to try larger bowls and platters, spindle work and combinations of these which may drive you to replace your original purchases.
If you are 'space limited' in your workshop I suggest you buy the largest lathe possible, (up to a 30" between centres capability). Go for a lathe with a swivelling headstock to allow you to turn larger items to the side of the bed. If you can stretch the budget, get a lathe with variable speed.
Tools: Save initial outlay by buying only the tools you need, but buy good quality items. To turn small bowls you will only need a bowl gouge, (3/8" would be my choice but a 1/4" will be OK if you stick to small bowls), and a round nose scraper, (1/2" will be fine). Add a small parting tool and you will be able to decorate your bowls. I'm a great advocate of Robert Sorby tools but Ashley Isles, Hamlet, Record Crown etc. are all fine - go for High Speed Steel, (HSS), tooling. The Proxon tools you have mentioned are not suitable for small bowls - these tools are designed for miniature turning - I suggest you buy full sized tools.
Chucks: These are expensive items which are really a matter of personal taste, look at Sorby Patriot, Axminster Evolution, Toolpost Versa and Record Supernova 2. You can save money here by going for the Dakota XT700 which is a clone of the Record Supernova 2. All of these accept a wide range of jaw adaptors - not necessarily produced by their parent companies. Chucks need to be matched to your lathe's spindle nose thread, either by direct threading or an adaptor / back plate, consider the adaptability of the chuck purchased to future lathe upgrades.
Note: A chuck is not an essential item when starting to turn, a faceplate is normally supplied with a new lathe an can be used for a great deal of applications.
Finally, and I would suggest this be your first purchase, get a good book - if you haven't already read it, get Keith Rowley's "Woodturning A Foundation Course" - Don't buy anything until you have read this cover to cover.
Regards Mick