If individuals and small companies can also negotiate their own tax rates, you might be correct. Since they cannot, it is a subsidy. There is no difference between collecting X and giving Y back than collecting X - Y.
Let's say that a regional government has $100 million in tax revenue, and that they have a balanced budget. Some company is considering relocating to the region, which would turn a piece of property currently contributing $100,000 revenue, or close to that, into a piece of property plus corporate tax and personal income taxes contributing $1.2 million, but with an increase in required spending of another $100,000, for a gain in net revenue of $1 million.
Offering a tax break less than $1 million costs nothing today, but still means an increase for the regional government, just a smaller one.
Offering a subsidy means dipping into a currently-balanced budget today, and making it up, probably, a year from now.
If all projections work out as expected, then several years from now there's probably no difference. But this, there could be a cash flow difference immediately.
Also, it's pretty easy to tell existing companies: tax breaks are for new development that increase value, and the break is essentially an 80% (or whatever) discount on the new taxes. It's hard to tell existing companies: we give subsidies to new companies but we'll continue to take you for granted.
At least that's how I suspect existing companies in a region tend to hear "subsidies."