yo & dan, thank you for your input.
yo, yes, we did sign a contract and says that they do not return the deposit (that in my case it covers the whole logo design fee as the quotation/contract is tied together based on the way they work.). So, in my case, i have payed more than enough the up to now work so ethics regarding payments (from my side) are not an issue. To be honest though, i would appreciate if they are truly professional if they would partially return the deposit as they will not complete the logo design work.
As for your second point, as we are already in the second round, your suggestion is to give them a warning sign that we may not move on if I am not satisfied and not just out of the blue terminate the project?
Dan, i totally agree and i was/am aware of that and that is why i talked with 6 design studios/freelances that i pre-select and thought that may be good fit.
This is a good experience though for my future client management process as now i see things from a different perspective. Experience is not for free right ? You always have to give something whether is money, time or something else...