The T-shaped and L-shaped cable-ends of MagSafe-1 are completely interchangeable. The only problem comes if your computer used the later MagSafe-2, which is slimmer in overall height. For that situation, Apple sells a tiny US$10 adapter TO the Magsafe-2 shape. [From memory, MagSafe-2 was used in the 2015 MacBook Pro and others.]
Any Apple power adapter that provides the required Watts or MORE will work perfectly with your MacBook Pro, and cause no issues except one: It is heavier to carry than the 60W model. The Mac draws only what is needed from the power adapter, and even if you use an 85W adapter for a 60W 13-in MacBook, there are no possibilities for damage whatsoever. It behaves exactly the same.
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Using a "too small" power adapter can cause some issues, but not likely permanent damage. When a Mac is running with power adapter connected, it feels free to use all the power supplied and also feely 'borrows' power from the battery at the same time. Intensive computations can cause a temporary power shortage, and the Mac can spontaneously emergency power off because of that shortage.
The best use of a "too small" power adapter is for overnight charging with the Mac nominally OFF.