The answer is a big "depends".
Generally speaking the first meeting is exploratory and not that involved, because rarely are all the right players available at such a meeting or have time then and there for an in-depth requirements analysis. It is usually better that they get a list of homework items they need to collect and prepare for a subsequent meeting.
That said even if you could nail down the requirements in a first two-hour meeting, I would tend to chalk that up as a cost of sales and perhaps add those two hours to any contingency costs I'm proposing. Bottom line - you've got to win the business and the best way I have found to do that is to demonstrate competency. When the people who are doing the "selling" are also the same people who do the delivery (something I advocate strongly for integration shops) then the client gets this very warm feeling in side that "these folks really know what they are doing". You will certainly stand apart from any other firm bidding on the business using a more traditional sales-person and separate implementer approach.
But I don't always give this time away. If establishing the project agreement safely in fact requires substantial interaction and I am still in pre-sales mode and while I'm scoping out the opportunity among the many tasks I have on my plate to do early on in the business relationship (before any contract is signed) is looking for a way to make the project terms safer for all concerned - them and my firm.
Sometimes that means paid study. Sometimes for really large opportunities with lots of competition you have to do quite a bit of work unfunded. I used to bid on multi million dollar projects all the time where that was the case and I can tell you I did not much enjoy that.
If the project is sufficiently complex that it can't be safely scoped out in an exploratory meeting, or that the likely items that come up for change management could be costly to either party, I will instead sell them first on a more detailed requirements analysis and project planning phase.
You can do this without seeming like you merely want to get more of the fatted calf for yourself by simply positioning -- from the very start -- that every project has some distinct phases and the first of which is requirements analysis and planning.
Sometimes we can do sufficient needs gathering in a pre-sales meeting to safely conduct business, with an appropriate caution to the client that unexpected surprises have to go through a change management process where we will identify the newly identified problems and costs, and they will either agree to expand the scope of work or not or elect to stop the project in the tracks. Again you can position this as a positive along the way - the client has control over the process.
But often times the project is simply too complex, and/or the client is not nearly sophisticated enough to appreciate all the complexities, and more formal study and documentation of the situation, requirements and expected outcome is needed, and in this case you really ought to be charging for that work in one way or another, either by including these costs as part of your fixed price for deliverables, or billing them hourly / daily for the work effort in this phase.
If they are unable to understand the necessity of this, or are unwilling to fork up the cash to do the work properly in my experience they tend to be clients you don't really want. Fire them before you even get started and move on to the next opportunity.
Bottom line: structure your introductory pitch such that you are always talking about a multi-phased approach and that way if it becomes clear that substantial analysis work has to take place first, make it a win for them to buy your services to do that work and one of the deliverables at the end of that first phase will be proper costing for the rest. This is a supremely sane way of doing this business. Sadly not all clients are wise enough to proceed as such.
“Even those who arrange and design shrubberies are under
considerable economic stress at this period in history.”