Fulton Church of Christ

The same generic Hebrew word is used for peace offerings, the Passover lamb, burnt offerings, and sin offerings. The same word is used in Hosea 6:6 (quoted by Matthew 9:13, 12:7) and Psalm 40:6 (quoted by Hebrews 10:5, 8). No argument can be made based on which word was used. The context is the only clue concerning which type of sacrifice is referenced.
The same Greek word as in Matthew 9:13, 12:7, Hebrews 10:5, 8 is used in the New Testament to describe sin offerings (Hebrews 5:1, 7:27, 8:3, 9:9, 10:1, 10:3, 10:6, 10:11 ), Abel’s sacrifice (Hebrews 11:4). And temple sacrifices (Luke 13:1). Again, no argument can be made based on word choice. The context tells the type.
The same Greek word is used of:

Christians (Romans 12:1, Philippians 2:17)
Jesus (Ephesians 5:2, 1 Corinthians 5:7, Hebrews 9:23, 9:26, 10:12, 10:26)
A gift (Philippians 4:18)
Praise (Hebrews 13:15)
Doing good and sharing (Hebrews 13:16)
General Christian activities (1 Peter 2:5)

When an inspired writer is being poetic or wants to include all types of sacrifices, the phrase “sacrifice and offering” may be used (Hebrews 10:5, 8, quoting Psalm 40:6).
When Paul brought offerings for himself and four others, the exact type was not specified. If this was part of a Nazirite vow, the offerings were both sin and burnt offerings, perhaps including grain, drink, peace, wave, and heave offerings (Number 6:2 – 21). If the offerings were for some other type of uncleanness, the ritual would include sin and burnt offerings. There is not enough detail in Acts to establish a context. Offering is used in a generic sense.
The same Greek word as is used for grain offering (Hebrews 10:6, 8) is used for:

Yom Kippur (2 Corinthians 5:21)
Passover (1 Corinthians 5:7)
Peace and sin offering (Ephesians 5:2)
Cleansed heaven (Hebrews 9:23)
Blood atonement (Hebrews 9:25, 10:12)
Participation (1 Corinthians 10:16 – 21)
Reminder (1 Corinthians 11:23 – 27)

Evangelism (Romans 15:16)
Praise (Hebrews 13:15)
Doing good and sharing (Hebrews 13:16, Philippians 4:18)
General Christian activities (1 Peter 2:5)
Faith (Philippians 2:17)
Our bodies (Romans 12:1)

More details on the applications made in the New Testament of the Mosaic sacrificial system:

Specific comparisons are not made in the New Testament concerning heave or wave offerings.
Wave offerings are called such because they are waved before God, then given to the priest (Numbers 18:8 – 20). Heave offerings and wave offerings seem to be together. Usually the distinction is by weight (hard to wave a quarter of a calf), but at least once the heave offering is just a loaf of bread (Leviticus 7:14), but it could be a lot of loaves. In one place, only heave offerings are mentioned (Numbers 15:17 – 21), but I can see no distinction. Perhaps wave offerings are sheaves and heave offerings are products.

At Aaron’s ordination (Exodus 29:24 – 28, Leviticus 8:22 – 29)

Peace offering (Leviticus 7:11 – 34, 9:21)

Grain offerings (Leviticus 10:12 – 15)

Cleansing a leper (Leviticus 14:1 – 32)

Feast of Unleavened Bread, firstfruits (Leviticus 23: 6 – 15)

Feast of Pentecost, firstfruits (Leviticus 23:16 – 21)

Adultery test (Numbers 5:23 – 26)

Nazirite (Numbers 6:13 – 20)

Having no inspired commentary, all I can do is speculate. Wave and heave offerings are, in some cases, first fruits. Jesus, Christians, and the indwelling Spirit are all firstfruits.

Jesus is called a first fruit (1 Corinthians 15:20 – 23)

Christians who die before Judgment are called first fruits (James 1:18, Revelation 14:4)

First converts in a region (Romans 16:5, 1 Corinthians 16:15)

We have the first fruits of the Spirit (Romans 8:23)

The part of the sacrifice that became the portion of the priest was likened to gifts for those who spread the gospel (1 Corinthians 9:13 – 14)

Sacrifice