Do “100% renewable” electricity plans increase renewable electricity production? Well, it will depend on how the specific scheme is implemented, what the renewable suppliers are, and what timescale you are considering. So, the answer could be yes, no or maybe.
I don't know about your supplier's particular plan, and in order to keep this answer useful to as many readers as possible, I'll talk about the generalities that make a difference.
Most renewable sources have such low short-run marginal costs that they'll run pretty much any time, regardless of the current price in the market. So an increase in demand from consumers on renewable tariffs is very unlikely to result in immediate increase in renewable supply. There might be corner cases on small grids where the marginal producer is a renewable supplier with non-trivial short-run marginal costs, such as biomass or biogas: in such cases, increased demand from people on 100% renewable tariffs could cause an immediate and equivalent increase in renewable supply.
Most 100%-renewable tariffs tend to be based on the trading of certificates-of-origin of renewable electricity. Whether your tariff really makes a difference, will depend on whether or not there is a surplus of such certificates in the market.
Surplus: If there is a surplus, and it is expected that there will continue to be a surplus, (as is the case Britain with the REGOs - Renewable Energy Guarantees of Origin) then no, the tariff makes no difference at all. The situation in this case is exactly as you describe: "your" electricity becomes 100% renewable, and the electricity of those on the "conventional" plan has a higher proportion of non-renewable energy than before.
No surplus: If there is not a surplus of such certificates now, or there is expected not to be one over the next few years, then the price of those certificates will rise, incentivising investment in new renewable generation infrastructure. The increase in renewable generation might be more or less than the amount you have consumed.
Some companies (such as Ecotricity in the UK) choose to invest a large proportion of their revenue into new renewable capacity. So buying from a company such as them, does increase renewable electricity production, over the scale of years, regardless of what's going on with the generation-specific and scheme-specific issues set out above. In such cases, the increase in renewable generation tends to be much more than the amount you have consumed.