From PMI point of view -

1) estimate costs is about estimating costs for the activities. The outputs are activity cost estimates and basis of estimates.

2) determine budget is adding all those costs or rolling up the lower level costs to the highest level cost. You add contingency reserves to it and phase it over the project shcedule. The output is the cost performance baseline. Other output is project funding requirement which is as per the funding requirement and the funding limit reconciliation. I.e., 20K in Q1, 30 K in Q2 etc.

Please remeber the cost performance baseline does not include the management reserves.

Example -

1) You can estimate costs for activities or the work pacakages or higher level like design cost, built cost, testing cost. Within Design you may have breakup like high level design, infrastructure design, low level component design etc and you can estimate at these level before rolling up.

2) You need to add all the above and contingency reserves and create a time phased cost performance baseline.